Method and apparatus for reserve measurement

ABSTRACT

The present invention describes a method and apparatus for constructing a historically based frequency distribution of unknown ultimate outcomes in a data set, the method comprising the acts of: (A) collecting relevant data about a series of known cohorts, where a new group of the data emerges at regular time intervals, measuring a characteristic of each group of the data at regular time intervals, and entering each said characteristic into a data set having at least two dimensions; (B) determining a number of frequency intervals N to be used to construct said distribution of unknown ultimate outcomes; (C) for each period I, constructing an aggregate distribution by: (a) calculating period-to-period ratios of the data characteristics; (b) identifying a range of ratio outcomes for cohort I; (c) constructing subintervals for cohort I; and (d) calculating all possible ratio outcomes for cohort I; and (D) constructing a convolution distribution of outcomes for all said possible ratio cohorts combined, by: (a) selecting outcomes for any two cohorts A and B; (b) constructing a new range of outcomes for the convolution distribution of cohorts A and B; (c) constructing new subintervals for the convolution distribution of cohorts A and B; (d) calculating the combined outcomes for the two cohorts A and B to provide a resulting convolution distribution; and (e) combining the resulting convolution distribution with the distributions of outcomes for each remaining cohort by repeating each of the preceding acts D.(a) through D.(d) for each pair of cohorts.

TECHNICAL FIELD

The invention relates generally to methods for the determination ofhistorically based benchmarks against which estimates of future outcomesmay be compared, thus developing a measure of the reasonableness of suchestimates. More particularly, the invention develops historically basedbenchmarks against which estimates of property & casualty insurance lossreserves may be compared, thus developing a measure of thereasonableness of such loss reserve estimates.

BACKGROUND ART

In the property & casualty insurance (hereinafter “insurance”) industry,maintenance of proper loss and loss expense reserves (hereinafter “lossreserves”) is

-   -   (a) Legally required,    -   (b) A vital element in the determination of the financial        condition of an insurance company, and    -   (c) A major determinant of the current income and associated        income statements.

On one hand, over the years, a large variety of methodologies have beendeveloped for the determination of estimates of loss reserves. On theother hand, there has been a virtual vacuum in the area ofidentification of historical benchmarks against which such loss reserveestimates may be compared, thereby providing a means for thedetermination of the reasonableness of such loss reserve estimates.

The process of estimating insurance company reserves involves fourprimary elements: raw data, assumptions, methods of estimation, andjudgment of the loss reserve specialist (e.g., an actuary). Thus thevarious estimates that a loss reserve specialist makes necessarily relyon the judgment of the loss reserve specialist in the selection ofassumptions and methods and ultimately in making the final reserveselection. While the application of judgment is an indispensable elementin the process of arriving at loss reserve estimates, the manner ofassessing the reasonableness of such estimates (via the identificationof historically based benchmarks) remains a largely unexplored subject.It would be useful to have objective historically based benchmarksagainst which loss reserve estimates may be compared.

One direct method for developing such objective historically basedbenchmarks involves the use of historical ratios generated by comparingconsecutive valuations of various cohorts of losses (e.g., lossesincurred during a particular year or other time period) as they developfrom one time period to another. To identify a historically basedbenchmark for loss reserve estimates, one can calculate period to periodratios for known consecutive valuations of cohorts of losses and usecombinations of such ratios to project outcomes for all the cohorts forwhich future valuations have yet to emerge. The collection of all suchoutcomes forms an empirical frequency distribution of all the possibleoutcomes with all the statistical measures associated with a frequencydistribution (such as mean, standard deviation, variance, and mode.)These statistical measures provide useful tools for assessing thereasonableness of loss reserve estimates.

Unfortunately, while this direct method can identify every possibleoutcome based on the application of historical valuation-to-valuationratios (i.e., possible “actual” outcomes), in practice the number ofpossible outcomes becomes unwieldy for even fairly small data sets. Forlarger data sets (i.e., involving more than ten cohorts), the process ofcalculating all possible outcomes becomes impractical, because of thedramatic increase in the amount of computing power necessary tocalculate all possible outcomes.

An indirect solution exists. Instead of using calculated outcomes,individual outcomes for any one cohort can be slotted as they arecalculated for each cohort (such as all losses incurred in a specifictime period) into a set of N intervals, with N sufficiently large suchthat the difference between any calculated outcome and its surrogate(the midpoint of the appropriate interval) is not more than any givendegree of tolerance, ε. For our purposes ε is expressed as a percenttolerance. In other words, a calculated outcome is never more than ε%from its surrogate. Once the N intervals are set for each cohort foreach line of business, there will be N distinct outcomes for eachaccident year for each line of business (each outcome being representedby the midpoint of an interval), and each distinct outcome having anassociated frequency (The frequency associated with a specific midpointis equal to the number of times a true calculated possible outcome isslotted in that interval). These individual distributions (one for eachcohort, and each consisting of N distinct outcomes, with each distinctoutcome having an associated frequency) are then combined to produce yetanother distribution that combines all cohorts (accident years) and alllines of business. This convolution distribution is the underlyingdistribution that is implied by the given data arrays. It may be used tocalculate a wide assortment of probabilities for various reservingpropositions; and thus enable the development of a substantial measureof the reasonableness of any given loss reserve estimate.

DISCLOSURE OF INVENTION BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings illustrate a complete exemplary embodiment ofthe invention according to the best modes so far devised for thepractical application of the principles thereof, and in which:

FIG. 1 illustrates an exemplary manner in which a subinterval isconstructed so as to observe the error tolerance.

FIG. 2 illustrates an exemplary manner in which the sum of twosubintervals, each of which meets the error criterion, also meets theerror criterion.

FIG. 3A shows a graph of an exemplary convolution distribution for twosample data sets (shown as Tables A and B).

FIG. 3B shows the graph of an exemplary basic distribution produced forTable A.

FIG. 3C shows the graph of an exemplary basic distribution produced forTable B.

FIG. 4 shows a flow chart for an exemplary process according to theinvention.

TABLE A. Sample Data Set A.

TABLE B. Sample Data Set B.

TABLE C. Shows tabular distribution of outcomes associated with Table A.

TABLE D. Shows tabular distribution of outcomes associated with Table B.

TABLE E. Shows tabular distribution of outcomes that represent theconvolution of distributions shown in Tables C and D.

APPENDIX A. This is the basic program that produces Tables C and D forData Sets A and B.

APPENDIX B. This is the convolution program that takes Tables C and Dand combines them into Table E and Drawing 3A.

BEST MODE(S) FOR CARRYING OUT THE INVENTION

In a preferred embodiment, a process for calculating distributionoutcomes is provided. This process can be implemented, for example, by acomputer program, by electronic hardware specifically designed toexecute the process or software implementing the process, by amicroprocessor storing firmware instructions designed to cause computerhardware to carry out the process, or by any other combination or hybridof hardware and software. The process can also be embodied in a computerreadable medium that can be executed by computer hardware or software toimplement the disclosed process.

Assumptions

A. It is assumed that data will be provided for a number of lines ofbusiness K. Thus K=1, 2, 3, . . . , k, . . . , K−2,K−1,K.

B. It is assumed that each line of business has a historical databasefor I accident years. Thus I=1, 2, 3, . . . i, . . . , I−2, I−1, I. Themost mature (oldest) year is designated year 1.

C. It is assumed that each accident year is developed through J periodsof development. Thus J=1, 2, 3, . . . , j, . . . , J−2, J−1, J.

D. It is assumed that I≧J (i.e. that no accident year develops longerthan the total number of years in the historical database). Thisassumption allows one to cut off the loss development after a number ofyears have passed, as is done in Schedule P filed by insurance companieswith the state regulatory authorities. (Schedule P is a series ofexhibits required to be included in the Statutory Financial Statementsof insurance companies in which, for each line of business and for alllines combined, each accident year is valued at annual intervals for amaximum of ten years of development. In other words, the tracking ofvaluation of individual accident years is abandoned after ten years onthe premise that the vast majority of loss values have emerged by thattime.)

Calculation of N

First, the user determines the number of intervals N needed such thateach calculated outcome is no more than a given percent tolerance ε fromits slotted value at the midpoint of an interval.

1. The degree of tolerance, ε, is determined by the user.

2. The user also makes use of the ultimate valuation for accident years1 through J as of the end of J years of development. Such valuationsafter J years have passed are routinely provided by insurance companies,on an annual basis, to the regulatory authorities. In other words, theprocess makes use of the historical factors utilized by the insurancecompany for the purpose of making an ultimate estimate for a cohort ofclaims after the required ten years of tracking has expired.

3. Each valuation point is designated by given V_(i,j,k), where i is theaccident year, j is the year of development, and k is the line ofbusiness. Thus V_(2,3,6) represents the value associated with accidentyear No. 2, at the end of development period No. 3, for line of businessNo. 6.

4. Finally, a loss development factor is defined as the ratio of thevaluation at time j+1 to the value at time j, orL_(i,j,k)=V_(i,j+1,k)/V_(i,j,k).

Thus, the data needed to drive the process would appear in an arraysimilar to the following (this example shows only line of business No.1—and other arrays would be provided for the remaining lines ofbusiness): AY 1 2 3 . . . j . . . J − 2 J − 1 J ∞ 1 V_(1,1,1) V_(1,2,1)V_(1,3,1) . . . V_(1,j,1) . . . V_(1,J−2,1) V_(1,J−1,1) V_(1,J,1)V_(1,∞,1) 2 V_(2,1,1) V_(2,2,1) V_(2,3,1) . . . V_(2,j,1) . . .V_(2,J−2,1) V_(2,J−1,1) V_(2,J,1) V_(2,∞,1) 3 V_(3,1,1) V_(3,2,1)V_(3,3,1) . . . V_(3,j,1) . . . V_(3,J−2,1) V_(3,J−1,1) V_(3,J,1)V_(3,∞,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . i V_(i,1,1) V_(i,2,1) V_(i,3,1) . . . V_(i,j,1) . . . V_(i,J−2,1)V_(i,J−1,1) V_(i,J,1) V_(i,∞,1) . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . J V_(J,1,1) V_(J,2,1) V_(J,3,1) . . .V_(J,j,1) . . . V_(J,J−2,1) V_(J,J−1,1) V_(J,J,1) V_(J,∞,1) . . . . . .. . . . . . I − 2 V_(I−2,1,1) V_(I−2,2,1) V_(I−2,3,1) I − 1 V_(I−1,1,1)V_(I−1,2,1) I V_(I,1,1)A. Constructing N for accident year I for line of business No. 1, orconstructing N_(I,1).

Constructing the maximum and minimum loss development factors for eachdevelopment period. For each development period, all loss developmentfactors are identified, and then the maximum (Max) and minimum (Min)loss factors are identified for each such set. For example, for year i,the set of loss development factors through two years of developmentconsists of all Loss Development Factors of the form L_(i,1,1), or{L_(1,1,1); L_(2,1,1); . . . ; L_(i,1,1); . . . ; L_(I−2,1,1);L_(I−1,1,1)}. The Max and Min of this set is denoted by: Max {L_(i,1,1)}and Min {L_(i,1,1)}, both taken over the index i, respectively; i=1, 2,3, . . . , I−1. This process is repeated for each development period.This results in a set of maximums and minimums of the form Max{L_(i,j,1)} and Min {L_(i,j,1)}, with each development period yielding amax and a min loss development factor.

Constructing the maximum and minimum values for the cumulative lossdevelopment factors. Having identified the maximum and minimum lossdevelopment factor for each development period, now the max and mincumulative loss development factors for accident year I are constructedby multiplying together all the max and all the min loss developmentfactors. For example:

Max cumulative loss development factor=II (Max {L_(i,j,1)}), with the“Max function” ranging over i and the “II function” ranging over j.

Min cumulative loss development factor=II (Min {L_(i,j,1)}), with the“Min function” ranging over i and the “II function” ranging over j.

Thus, the difference between the maximum and minimum values of alloutcomes for all products of loss development factors for year I isgiven by the quantity:[II (Max {L_(i,j,1)})−II (Min {L_(i,j,1)})].

Any specific ultimate outcome for year I must fall somewhere along theclosed interval defined by:[II (Min {L_(i,j,1)}), II (Max {L_(i,j,1)})].

Constructing the subintervals. The goal is to determine the numberN_(I,1), a number of subintervals for year I, such that (a) if theinterval containing the full range of outcomes is divided into thesesubintervals, and (b) any calculated value that falls in thatsubinterval is replaced with the midpoint of that subinterval, then (c)the true (computed) value cannot be more that ε away from the midpointof that subinterval.

The target number is denoted by N_(I,1). The interval[II (Min {L_(i,j,1)}), II (Max {L_(i,j,1)})]is divided into (N_(I,1)−1) equal subintervals. The width of any one ofthe new subintervals is given by:[II (Max {L_(i,j,1)})−II (Min {L_(i,j,1)})]/(N_(I,1)−1)and the radius of each subinterval is defined as one-half that number,or:[II (Max {L_(i,j,1)})−II (Min {L_(imj,1)})]/2(N_(I,1)−1).

In practice, the subinterval can be open or closed on either end, tosuit the particular application. For convenience, the subintervaldefined here is an open/closed subinterval, with the leftmost pointbeing excluded from the subinterval and the rightmost point beingincluded in the subinterval. The leftmost point of the fall range [thatis, II (Min {L_(i,j,1)})] is designated as the midpoint of the firstsubinterval. Then the full leftmost subinterval is given by:[II (Min {L_(i,j,1)})−[(II (Max {L_(i,j,1)})−II (Min{L_(i,j,1)})]/2(N_(I,1)−1)], II (Min {L_(i,j,1)})+[[II (Max{L_(i,j,1)})−II (Min {L_(i,j,1)})]/2(N_(I,1)−1) ]].

The rightmost subinterval is similarly defined and is given by:[II (Max {L_(i,j,1)})−[[II (Max {L_(i,j,1)})−II (Min{L_(i,j,1)})]/2(N_(I,1)−1)], II (Max {L_(i,j,1)})+[[II (Max{L_(i,j,1)})−II (Min {L_(i,j,1)})]/2(N_(I,1)−1)]].

This particular construction restores the odd subinterval that wassubtracted from N_(I,1) to arrive at the width of a subinterval.

Meeting the tolerance criterion, solving for N_(I,1). The number ofsubintervals, N_(I,1), that will assure tolerance criterion ε is met arenow calculated.

Once a true value has been placed in its appropriate subinterval, itcannot be more than the radius of the subinterval away from its proposedsurrogate (the midpoint of that subinterval). Thus the maximum error isthe radius of the subinterval constructed above:[II(Max {L_(i,j,1)})−II(Min {L_(i,j,1)})]/2(N_(I,1)−1).

Thus the true error (the distance from the true value to the midpoint ofthe associated subinterval) is always less than or equal to the maximumerror (the radius of the subinterval as given above. So instead ofdealing with the true error, a more stringent requirement is imposed,that the ratio of the radius of the subinterval to the midpoint of thesubinterval be less than ε. In other words:{[II(Max {L _(i,j,1)})−II(Min {L _(i,j,1)})]/2(N _(I,1)−1)}/Midpoint ofsubinterval<ε.

Now note that:{[II (Max {L _(i,j,1)})−II (Min {L _(i,j,1)})]/2(N _(I,1)−1)}/Midpointof subinterval≦{[II (Max {L _(i,j,1)})−II (Min {L _(i,j,1)})]/2(N_(I,1)−1)}/II(Min{L _(i,j,1)})since the “midpoint of the subinterval” is at least equal to or greaterthan II (Min {L_(i,j,1)}).

Thus the tolerance condition is met if N_(I,1) is selected such that:{[II (Max {L _(i,j,1)})−II (Min {L _(i,j,1)})]/2(N _(I,1)−1)}/II(Min {L_(i,j,1)})<ε.

Solving for N_(I,1) one obtains:N _(i,1)>1+(½ε)[ II (Max {L _(i,j,1)})−II (Min {L _(i,j,1)})]/II (Min {L_(i,j,1)}).

The value N_(I,1) is therefore sufficient so that when each true,computed value is replaced with the midpoint of the appropriatesubinterval, the true value is never more than ε away from itssurrogate, the midpoint of the subinterval.

B. Constructing N for line of business 1, or N_(I).

Having constructed N_(I,1), the process is repeated as often asnecessary to construct a corresponding N value for each accident year tobe projected to ultimate, thus yielding an entire set of N values forline of business No. 1:N_(I,1); N_(I−1); N_(I−2,1); N_(I−3,1); . . . ; N_(J+2,1); N_(J+1).

For each of these N values, the true value is never more than ε awayfrom the midpoint of the corresponding subinterval for each accidentyear, from accident year J+1 to accident year I. The maximum of allthese N_(i,1) values is selected to ensure that this condition (of theerror being less than ε) is met for every single accident yearindividually. Thus, instead of a set of N_(i,1) values, Max {N_(i,1)} isused, with i ranging from J+1 to I. This value is designated N₁, meaningthe N value associated with line of business No. 1.

C. Constructing N for all lines of business.

Once N₁, N₂, N₃, . . . , N_(K), have been constructed, the maximum ofthese N values, Max {N_(i), i=1,2,3, . . . , k, . . . , K}, is selected,so that maximum N is sufficient to satisfy the ε criterion for everysingle line of business.

Although this exemplary embodiment employs the above method for thecalculation of N, N may also be a number chosen arbitrarily by the user,or may be based upon other considerations, such as, for example, themaximum number of intervals that could be calculated within a givenamount of time by the computer used by the user to execute the program,or some given number that is high enough that ε is sufficiently low forthe user's purposes regardless of the particular characteristics of thedataset to be evaluated (for example, if the N that provides a givenerror level ε is virtually always between 500 and 600, a user couldselect N=1000 rather than calculate N for each dataset). In the eventthat N is determined to meet some other criteria, it is still necessaryto provide the historical loss data for each accident year for each lineof business. Note also that when N is determined by other criteria,there is no assurance that the error tolerance ε is met. The processdescribed below requires that the original data array has been providedregardless of whether or not it is used to determine N.

Construction of the Convolution Distribution

Once N is determined, and N and the valuations described above have beenprovided, for example, entered as a value in a computer program, theprocess proceeds as follows:

A. Constructing the aggregate loss distribution for one year, and forthis illustration accident year I.

The process consists of the following actions:

1. Identifying the range of outcomes for accident year I. Using theMax/Min functions described above, the Max/Min cumulative lossdevelopment factors are calculated, and those are multiplied by thelatest valuation available for the accident year I. Thus the Max/Minultimate values for accident year I are determined.

2. Constructing the subintervals for accident year I. Given the Max/Minultimate values for accident year I, the N subintervals described aboveare identified.

3. Calculating all the different outcomes for accident year I. Asdiscussed above, the product of each combination of loss developmentfactors and the latest valuation for accident year I is calculated. Aseach outcome is calculated, the interval in which it belongs isdetermined and the outcome is replaced with the midpoint of thatinterval, and the frequency of outcomes appearing in that interval isincreased by 1. The process continues until all combinations arecalculated and all possible outcomes have been determined for accidentyear I. All results are slotted and their frequency is calculated.

This process creates an aggregate loss (frequency) distribution foraccident year I.

B. Constructing the aggregate loss distribution for each of theremaining accident years.

The process described in Section A above for accident year I is thenrepeated for each of the remaining accident years. This results in a setof individual aggregate loss distributions, one for each accident year,and each consisting of N intervals, with each interval having anassociated frequency.

C. Creating the convolution distribution for all accident years combinedwithin one line of business.

This process consists of the following actions:

1. Selecting two accident years from the set of all open accident years.Select any two accident years, preferably starting with the two mostmature years.

2. Creating the new range of outcomes for the convolution distributionof the two accident years. This task is accomplished by calculating (a)the sum of the two greatest midpoints of the two component distributionsand (b) the sum of the two smallest midpoints of the componentdistributions. These calculations result in a new Max/Min for the twoaccident years combined.

3. Creating the new subintervals for the convolution distribution of thetwo selected accident years. Once again, divide the new interval into Nsubintervals as described above.

4. Calculating the combined outcomes for the two accident years. Everyoutcome from the first component distribution is then added to everyoutcome of the second component distribution, and the results areslotted in the new N subintervals constructed in the prior step. Thefrequencies for each two subintervals thus added are multiplied andtagged as belonging with the combined subinterval. This process yieldsthe first convolution distribution—the one belonging to the two selectedaccident years.

5. Creating the ultimate convolution distribution for all accident yearsfor a line of business. Actions 1-4 are then repeated; combining thefirst convolution distribution derived in step 4 immediately above withthe distribution of outcomes of another accident year. This processyields a second convolution distribution representing the combineddistribution for the three selected accident years. The process isrepeated until all accident year outcomes have been combined.

The result is a single aggregate (convolution) loss distribution for aline of business.

D. Creating the convolution distribution for all lines of businesscombined.

This process consists of Steps 1-5 as described in the immediatelypreceding section except that the component distributions are thosebelonging to lines of business. The end result is an aggregate(convolution) loss distribution for all lines of business combined, forthe given insurance company.

The above described method may be implemented by instructions stored ona “computer readable medium.” The term “computer readable medium” asdescribed herein refers to any medium that participates in providinginstructions to a computer processor for execution. Such a medium maytake many forms, including, but not limited to, non-volatile media,volatile media, and transmission media Non-volatile media include, forexample, optical or magnetic disks. Volatile media include dynamicmemory, such as the random access memory (RAM) found in personalcomputers. Transmission media may include coaxial cables, copper wire,and fiber optics. Transmission media may also take the form of acousticor light (electromagnetic) waves, such as those generated during radiofrequency (RF) and infrared (IR) data communications. Common forms ofcomputer readable media include, for example, a floppy disk, a harddisk, magnetic tape, CD-ROM, DVD-ROM, punch cards, paper tape, any otherphysical medium with patters of holes, RAM, PROM, EPROM, FLASHEPROM,other memory chips or cartridges, a carrier wave, or any other mediumfrom which a computer can read instructions.

The present invention has been described in sufficient detail to teachits practice by one of ordinary skill in the art. However, the abovedescription and drawings of exemplary embodiments are only illustrativeof preferred embodiments that achieve the objects, features andadvantages of the present invention, and it is not intended that thepresent invention be limited thereto. Any modification of the presentinvention that comes within the spirit and scope of the following claimsis considered part of the present invention.

INDUSTRIAL APPLICABILITY

The present invention has utility, for example, in the property andcasualty insurance industry, to assist in satisfying legal requirementsin the field, and to efficiently determine estmates of loss reservesnecessary to conduct business.

ANNEX 1 Demonstration of Validity of N as Calculated

A. Demonstrating that the ε condition remains satisfied when thecumulative loss development factors are applied to a base number (thegiven, and latest, value).

All work thus far has been performed for just the cumulative lossdevelopment factors. In reality, when one projects ultimate values, onetakes the cumulative loss development factor and multiplies it by thelatest reported value. When all the calculations carried out above arecarried out with this last step included (i.e., multiplying the latestvalue by the cumulative loss development factor), it will be readilyseen that the latest reported amount simply cancels out at all points ofthe calculation. For example, if we take the final formula for N_(I,1)developed above, we have:N _(I,1)>1+(½ε)[II (Max {L _(i,j,1)})−II (Min {L _(i,j,1)})]/II(Min {L_(i,j,1)}).

And if each cumulative loss development factor is multiplied by therelevant latest reported value, V_(I,1,1), we would have:N _(I,1)>1+(½ε) [(V _(I,1,1)) II (Max {L _(i,j,1)})−(V _(I,1,1)) II (Min{L _(i,j,1)})]/(V _(I,1,1)) II(Min {L _(i,j,1)}).

And V_(I,1,1) cancels out from all parts of the major fraction. And thesame is true for all other accident years.

B. Demonstrating that the ε condition remains satisfied whenaccidentyears are combined (i.e., added) in order to arrive at theaggregate loss distribution for all accident years combined, all withinline of business No.1.

Observation. Given two sets of intervals, each set consisting of nsubintervals of identical width, one set spanning the interval (a−Δ₁,a+(2n−1)Δ₁), where Δ₁ is the radius of a subinterval, that has themidpoints of the component intervals placed at a+2iΔ₁, with i rangingfrom 0 to n−1, and the other set spanning (b−Δ₂, b+2n−1)Δ₂), where Δ₂ isthe radius of a subinterval, that has the midpoints of the respectiveintervals placed atb+2iΔ₂, with i ranging from 0 to n−1, one can thenconstruct a new set of subintervals consisting of the “sum” of the twooriginal sets of intervals, spanning ((a+b)−(Δ₁+Δ₂),(a+b)+(2n−1)(Δ₁+Δ₂)), each having a with of (Δ₁+Δ₂).

The midpoints of the new set of subintervals would be located at(a+b),(a+b)+2(Δ₁+Δ₂), (a+b)+4(Δ₁+Δ₂), . . . , (a+b)+2(n−1)(Δ₁+Δ₂). Andthus the radius of the new subintervals (i.e., Δ₁+Δ₂) would be equal tothe sum of the radii of the two component subintervals.

With this background, let us now consider two sets of subintervals, witheach set consisting of n subintervals, with the subintervals havingradii of Δ₁ and Δ₂, for the two sets, respectively, with the midpointsof the respective sets of subintervals given as follows:Set A: a, a+2Δ₁, a+4Δ₁, a+6Δ₁, a+8Δ₁, a+10Δ₁, . . . , a+2(n−1)Δ₁Set B: b, b+2Δ₂, b+4Δ₂, b+6Δ₂, b+8Δ₂, b+10Δ₂, . . . , b+2(n−1)Δ₂

Let us now assume that Set A is the set of subintervals produced forCohort A, consisting of a group of losses (e.g., the losses incurredduring a specific accident year) and that Set B is the set ofsubintervals produced for Cohort B, consisting of another group oflosses (e.g., the losses incurred during another specific accident year)By our construction thus far, we know that any true calculated value ofultimate outcomes produced for Cohort A has been replaced by one of themidpoints associated with Set A. We constructed these subintervals suchthat the error generated by substituting a true calculated value with amidpoint of a subinterval is not greater than ε. Put yet differently,the difference between any true calculated value V_(a) and the nearestmidpoint of the subintervals in Set A is not more than Δ₁. Therefore,the ratio of Δ₁ to the leftmost point of all the subintervals in Set A,that is (a−Δ₁), is less than ε. In formula form this is given by:Δ₁/(a−Δ ₁)<ε

Similarly, for Set B, we can reach the conclusion that a true calculatedvalue V_(b) meets the following parallel construction noted above forSet A.:Δ₂/(b−Δ ₂)<ε

Given that if one had infinite computing power, one would never resortto substituting midpoints of subintervals for true calculated values, itis appropriate at this point to inquire about the amount of error thatone generates by adding two surrogates (midpoints) for V_(a) and V_(b),when both V_(a) and V_(b) individually meet the error tolerancecriterion ε. Thus the question becomes: what can be said about(Δ₁+Δ₂)/[(a−Δ₁)+(b−Δ₂)]in relation to the original error tolerance ε?

The tolerance condition Δ₁/(a−Δ₁)<ε implies that Δ₁<(a−Δ₁)ε.

Similarly, the tolerance condition Δ₂/(b−Δ₂)<ε implies that Δ₂<(b−Δ₂)ε.Adding the two inequalities yields:(Δ₁+Δ₂)<[(a−Δ ₁)ε]+[(b−Δ ₂)ε]or:(Δ₁+Δ₂)<[(a−Δ ₁)+[(b−Δ ₂)]ε

Dividing both sides of the inequality by [(a−Δ₁)+(b−Δ₂)] yields:(Δ₁+Δ₂)/[(a−Δ ₁)+(b−Δ ₂)]<ε

Thus when adding one accident year's approximation to another's, wheneach approximation meets the ε condition, it is demonstrated that thesum of the two approximations also meets the ε condition. And, this kindof demonstration can continue to be extended, one cohort at a time,until all the cohorts in a data array have been accounted for.

C. Demonstrating that the ε condition remains satisfied when aggregatedistributions for two lines of business are added together.

Using the identical logic as that used above in Section B, it ispossible to demonstrate that when two distributions of outcomes, each ofwhich meeting the ε criterion, will continue to meet the ε criterionwhen the convolution distribution is constructed by adding therespective outcomes from each of the two distributions.

DRAWING NO. 1 Illustration of the manner in which a Subinterval isConstructed such that the Error Tolerance is Met

The line segment (a,b) represents a typical subinterval, having amidpoint at M (=½(a+b)), such that a calculated point, such as x, may beslotted in this subinterval, and x is ultimately replaced by m.

The interval (A,B) is the segment bounded by A, the smallest midpoint ofall subintervals, and B, the largest midpoint of all subintervals. Thusthe midpoints of all subintervals are evenly spaced within this largerinterval.

The point corresponding to x designates a typical calculated outcome. Inthis illustration it is selected to between the midpoint M and theendpoint b.

The true error that is generated by replacing x with m is given by theamount |x−m|.

The maximum error that is possible is denoted by Δ=|m−b|.

Requiring that the replacement of x by m does not generate an errorgreater than ε means requiring that the error is less than the ratio of|x−m|/m.

In the construction advanced by this invention we assure this conditionis met by going through the following transformation:ε=|x−m|/m≦|m−b|/m≦|m−b|/A  (1)

Thus dividing (A,B) into sufficiently large number of subintervals suchthat the condition in (1) is met assures that the subintervalconstruction preserves the accuracy requirement.

DRAWING NO. 2 Illustration of the manner in which the sum of twoIntervals, each of which meets the Error Criterion, also meets the ErrorCriterion

Given a subinterval from the set of subintervals produced for Cohort Isuch that the subinterval construction meets the error criterion ε:

And given a subinterval from the set of subintervals produced for CohortII such that the subinterval construction meets the error criterion ε:

Then the construction of the combination of these two subinterval into anew subinterval (thus forming a convolution subinterval) yields thefollowing:

The error that would be generated if x+x′ was replaced with m+m′ isgiven by:|(x+x′)−(m+m′)|

And we wish for this amount to be less than the specified tolerance ε.

Thus we construct the following sequence of successively more stringentconstraints:|(x+x′)−(m+m′)≦|(b+b′)−(m+m′)|/(m+m′)≦|(b+b′)−(m+m′)|/(a+a′)|  (2)

We already have, by construction, the conditions that|b−m|/a<ε|and |b′−m′|/a′<ε.

Or, equivalently,|b−m|<aε|and |b′−m′|<a′ε.

Adding both sides of the inequalities yields:|b−m|+|b′−m′|<aε+a′ε=(a+a′)ε.

Dividing both sides by (a+a′) yields the desired condition as shown in(2) above. SAMPLE DATASET A Calculation of N for Set A Given: Errortolerance not more than 1/10 of 1% Calculate N so that the 1/10 at 1%condition is met Raw Data Valued After Indicated Number of Years Year 12 3 4 5 6 7 8 9 10 Ultimate 1988   1985974   2287598   2354278   2371587  2357456   2355474   2350474   2348454   2348442   2348442   23484421989   2049857   2384957   2484954   2501458   2501047   2564584  2560028   2554898   2554861   2554861   2554544 1990   2154154  2557450   2615487   2340582   2340005   2348591   2345888   2344482  2344824   2344824   2344824 1991   2356475   2758745   2805745  2826475   2846587   2849856   2848858   2848954   2847861   2854675  2847861 1992 2,495,542 2,831,272 2,867,221 2,891,206 2,967,7732,973,418 2,911,341 2,858,341 2,858,341 2,858,341 2,858,341 19932,872,429 3,518,051 3,609,053 3,602,762 3,553,539 3,541,706 3,469,9453,477,817 3,477,817 1994 3,428,730 4,078,597 4,446,098 4,445,6734,448,621 4,384,024 4,367,977 4,376,253 1995 2,839,386 3,680,2833,781,846 3,691,879 3,671,901 3,620,821 3,605,404 1996 2,685,8923,380,806 3,289,680 3,219,952 3,196,238 3,211,821 1997 3,039,8153,425,947 3,293,515 3,298,908 3,271,413 1998 3,132,568 3,760,6913,771,564 3,810,472 1999 4,402,008 5,574,428 5,853,402 2000 4,795,2615,613,134 2001 4,498,797 Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-1010-Ult. Period-to-period Link Ratios 1988 1.151877 1.029148 1.0073520.994042 0.999159 0.997877 0.999141 0.99999 1.00000 1.00000 19891.163475 1.041928 1.006642 0.999836 1.025404 0.998223 0.997996 0.999991.00000 0.99988 1990 1.187218 1.022693 0.894893 0.999753 1.0036690.998849 0.999401 1.00015 1.00000 1.00000 1991 1.170708 1.0170371.007388 1.007116 1.001148 0.999650 1.000034 0.99962 1.00239 0.997611992 1.134532 1.012697 1.008365 1.026483 1.001902 0.979123 0.9817951.00000 1.00000 1993 1.224765 1.025867 0.998257 0.986337 0.9966700.979738 1.002269 1.00000 1994 1.189536 1.090105 0.999904 1.0006630.985479 0.996340 1.001895 1995 1.296155 1.027597 0.976211 0.9945890.986089 0.995742 1996 1.258727 0.973046 0.978804 0.992635 1.004875 19971.127025 0.961344 1.001637 0.991665 1998 1.200514 1.002891 1.010316 19991.266338 1.050045 2000 1.170559 Max/Min Link Ratios Max 1.2961551.090105 1.010316 1.026483 1.025404 0.999650 1.002269 1.000146 1.0023931.000000 Min 1.127025 0.961344 0.894893 0.986337 0.985479 0.9791230.981795 0.999616 1.000000 0.997613 Cumulative Products at Max/Min LinkRatios Max 1.5092538 1.1644089 1.0681623 1.0572555 1.0299789 1.00446141.0048133 1.0025389 1.0023927 1.000000 Min 0.903463 0.8016354 0.83386910.9318083 0.9447155 0.9586356 0.9790761 0.9972303 0.997613 0.997613 N >1 + 1/(2 × 0.001) × [(1.50925) − (0.90346)]/(0.90346) = 336.261 Use N =337 or greater.

SAMPLE DATASET B Calculation of N for Set B Given: Error toelrance ofnotCalcualte N so that the 1/10 of more than 1/10 of 1% 1% condition is metRaw Data Valued After Indicated Number of Years Year 1 2 3 4 5 6 7 8 910 Ultimate 1988 325641 388932 380245 375214 385467 377826 377826 377024380458 381748 381748 1989 294758 355458 360452 380245 390245 401587401587 401587 401587 401587 401587 1990 350245 435142 429587 461523462536 465826 475826 475826 485745 485745 485745 1991 359848 429788409548 440526 440526 440526 444856 444856 444856 444900 444856 1992604287 660562 722626 810537 810537 845218 975537 960537 948037 948037948037 1993 282176 288093 314016 307709 307709 307709 301209 301209301209 1994 414267 502671 575367 618027 634806 606766 597266 587266 1995347207 345260 389196 389574 370421 367421 367421 1996 407584 425858498245 572172 643572 643572 1997 298564 356895 349158 345658 330958 1998674607 697101 705185 690264 1999 342252 414275 442215 2000 1149836 1277286  2001 596578 Year 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-UltPeriod-to-period Link Ratios 1988 1.19436 0.97766 0.98677 1.027330.98018 1.00000 0.99788 1.00911 1.00339 1.00000 1989 1.20593 1.014051.05491 1.02630 1.02906 1.00000 1.00000 1.00000 1.00000 1.00000 19901.24239 0.98723 1.07434 1.00219 1.00711 1.02147 1.00000 1.02085 1.000001.00000 1991 1.19436 0.95291 1.07564 1.00000 1.00000 1.00983 1.000001.00000 1.00010 0.99990 1992 1.09313 1.09396 1.12165 1.00000 1.042791.15418 0.98462 0.98699 1.00000 1993 1.02097 1.08998 0.97992 1.000001.00000 0.97888 1.00000 1.00000 1994 1.21340 1.14462 1.07414 1.027150.95583 0.98434 0.98326 1995 0.99439 1.12725 1.00097 0.95084 0.991901.00000 1996 1.04483 1.16998 1.14837 1.12479 1.00000 1997 1.195370.97832 0.98998 0.95747 1998 1.03334 1.01160 0.97884 1999 1.210441.06744 2000 1.11084 Max/Min Link Ratios Max 1.24239 1.16998 1.148371.12479 1.04279 1.15418 1.00000 1.02085 1.00339 1.00000 Min 0.994390.95291 0.97884 0.95084 0.95583 0.97888 0.98326 0.98699 1.00000 0.99990Cumulative Products of Max/Min Link Ratios Max 2.31469 1.86309 1.592411.38667 1.23282 1.18224 1.02431 1.02431 1.00339 1.00000 Min 0.800700.80521 0.84501 0.86327 0.90791 0.94987 0.97037 0.98689 0.99990 0.99990N > (1/(2 × 0.001) × [(4.01166) − (0.72452)]/(0.72452) = 945.427 Use N >946 Hence N should be anything greater than 946 (the max of 337 for setA and 946 for Set B) for two sets combined

Sample Data Set A Table of Outcome Intervals Outcome Intervals OutcomeIntervals From To As % Of All Outcomes −40,599,147 −40,554,5470.0000000000000% −40,554,545 −40,509,945 0.0000000000000% −40,509,944−40,465,344 0.0000000000000% −40,465,342 −40,420,742 0.0000000000000%−40,420,740 −40,376,140 0.0000000000000% −40,376,139 −40,331,5390.0000000000000% −40,331,537 −40,286,937 0.0000000000000% −40,286,935−40,242,335 0.0000000000000% −40,242,334 −40,197,734 0.0000000000000%−40,197,732 −40,153,132 0.0000000000000% −40,153,130 −40,108,5300.0000000000000% −40,108,529 −40,063,929 0.0000000000000% −40,063,927−40,019,327 0.0000000000000% −40,019,325 −39,974,725 0.0000000000000%−39,974,724 −39,930,124 0.0000000000000% −39,930,122 −39,885,5220.0000000000000% −39,885,520 −39,840,920 0.0000000000000% −39,840,919−39,796,319 0.0000000000000% −39,796,317 −39,751,717 0.0000000000000%−39,751,715 −39,707,115 0.0000000000000% −39,707,114 −39,662,5140.0000000000000% −39,662,512 −39,617,912 0.0000000000000% −39,617,910−39,573,310 0.0000000000000% −39,573,309 −39,528,709 0.0000000000000%−39,528,707 −39,484,107 0.0000000000000% −39,484,105 −39,439,5050.0000000000000% −39,439,504 −39,394,904 0.0000000000000% −39,394,902−39,350,302 0.0000000000000% −39,350,300 −39,305,700 0.0000000000000%−39,305,699 −39,261,099 0.0000000000000% −39,261,097 −39,216,4970.0000000000000% −39,216,495 −39,171,895 0.0000000000000% −39,171,894−39,127,294 0.0000000000000% −39,127,292 −39,082,692 0.0000000000000%−39,082,690 −39,038,090 0.0000000000000% −39,038,088 −38,993,4880.0000000000000% −38,993,487 −38,948,887 0.0000000000000% −38,948,885−38,904,285 0.0000000000000% −38,904,283 −38,859,683 0.0000000000000%−38,859,682 −38,815,082 0.0000000000000% −38,815,080 −38,770,4800.0000000000000% −38,770,478 −38,725,878 0.0000000000000% −38,725,877−38,681,277 0.0000000000000% −38,681,275 −38,636,675 0.0000000000000%−38,636,673 −38,592,073 0.0000000000000% −38,592,072 −38,547,4720.0000000000000% −38,547,470 −38,502,870 0.0000000000000% −38,502,868−38,458,268 0.0000000000000% −38,458,267 −38,413,667 0.0000000000000%−38,413,665 −38,369,065 0.0000000000000% −38,369,063 −38,324,4630.0000000000000% −38,324,462 −38,279,862 0.0000000000000% −38,279,860−38,235,260 0.0000000000000% −38,235,258 −38,190,658 0.0000000000000%−38,190,657 −38,146,057 0.0000000000000% −38,146,055 −38,101,4550.0000000000000% −38,101,453 −38,056,853 0.0000000000000% −38,056,852−38,012,252 0.0000000000000% −38,012,250 −37,967,650 0.0000000000000%−37,967,648 −37,923,048 0.0000000000000% −37,923,047 −37,878,4470.0000000000000% −37,878,445 −37,833,845 0.0000000000000% −37,833,843−37,789,243 0.0000000000000% −37,789,242 −37,744,642 0.0000000000000%−37,744,640 −37,700,040 0.0000000000000% −37,700,038 −37,655,4380.0000000000000% −37,655,437 −37,610,837 0.0000000000000% −37,610,835−37,566,235 0.0000000000000% −37,566,233 −37,521,633 0.0000000000000%−37,521,632 −37,477,032 0.0000000000000% −37,477,030 −37,432,4300.0000000000000% −37,432,428 −37,387,828 0.0000000000000% −37,387,827−37,343,227 0.0000000000000% −37,343,225 −37,298,625 0.0000000000000%−37,298,623 −37,254,023 0.0000000000000% −37,254,022 −37,209,4220.0000000000000% −37,209,420 −37,164,820 0.0000000000000% −37,164,818−37,120,218 0.0000000000000% −37,120,217 −37,075,617 0.0000000000000%−37,075,615 −37,031,015 0.0000000000000% −37,031,013 −36,986,4130.0000000000000% −36,986,411 −36,941,811 0.0000000000000% −36,941,810−36,897,210 0.0000000000000% −36,897,208 −36,852,608 0.0000000000000%−36,852,606 −36,808,006 0.0000000000000% −36,808,005 −36,763,4050.0000000000000% −36,763,403 −36,718,803 0.0000000000000% −36,718,801−36,674,201 0.0000000000000% −36,674,200 −36,629,600 0.0000000000000%−36,629,598 −36,584,998 0.0000000000000% −36,584,996 −36,540,3960.0000000000000% −36,540,395 −36,495,795 0.0000000000000% −36,495,793−36,451,193 0.0000000000000% −36,451,191 −36,406,591 0.0000000000000%−36,406,590 −36,361,990 0.0000000000000% −36,361,988 −36,317,3880.0000000000000% −36,317,386 −36,272,786 0.0000000000000% −36,272,785−36,228,185 0.0000000000000% −36,228,183 −36,183,583 0.0000000000000%−36,183,581 −36,138,981 0.0000000000000% −36,138,980 −36,094,3800.0000000000000% −36,094,378 −36,049,778 0.0000000000000% −36,049,776−36,005,176 0.0000000000000% −36,005,175 −35,960,575 0.0000000000000%−35,960,573 −35,915,973 0.0000000000000% −35,915,971 −35,871,3710.0000000000000% −35,871,370 −35,826,770 0.0000000000000% −35,826,768−35,782,168 0.0000000000000% −35,782,166 −35,737,566 0.0000000000000%−35,737,565 −35,692,965 0.0000000000000% −35,692,963 −35,648,3630.0000000000000% −35,648,361 −35,603,761 0.0000000000000% −35,603,760−35,559,160 0.0000000000000% −35,559,158 −35,514,558 0.0000000000000%−35,514,556 −35,469,956 0.0000000000000% −35,469,955 −35,425,3550.0000000000000% −35,425,353 −35,380,753 0.0000000000000% −35,380,751−35,336,151 0.0000000000000% −35,336,150 −35,291,550 0.0000000000000%−35,291,548 −35,246,948 0.0000000000000% −35,246,946 −35,202,3460.0000000000000% −35,202,345 −35,157,745 0.0000000000000% −35,157,743−35,113,143 0.0000000000000% −35,113,141 −35,068,541 0.0000000000000%−35,068,540 −35,023,940 0.0000000000000% −35,023,938 −34,979,3380.0000000000000% −34,979,336 −34,934,736 0.0000000000000% −34,934,734−34,890,134 0.0000000000000% −34,890,133 −34,845,533 0.0000000000000%−34,845,531 −34,800,931 0.0000000000000% −34,800,929 −34,756,3290.0000000000000% −34,756,328 −34,711,728 0.0000000000000% −34,711,726−34,667,126 0.0000000000000% −34,667,124 −34,622,524 0.0000000000000%−34,622,523 −34,577,923 0.0000000000000% −34,577,921 −34,533,3210.0000000000000% −34,533,319 −34,488,719 0.0000000000000% −34,488,718−34,444,118 0.0000000000000% −34,444,116 −34,399,516 0.0000000000000%−34,399,514 −34,354,914 0.0000000000000% −34,354,913 −34,310,3130.0000000000000% −34,310,311 −34,265,711 0.0000000000000% −34,265,709−34,221,109 0.0000000000000% −34,221,108 −34,176,508 0.0000000000000%−34,176,506 −34,131,906 0.0000000000000% −34,131,904 −34,087,3040.0000000000000% −34,087,303 −34,042,703 0.0000000000000% −34,042,701−33,998,101 0.0000000000000% −33,998,099 −33,953,499 0.0000000000000%−33,953,498 −33,908,898 0.0000000000000% −33,908,896 −33,864,2960.0000000000000% −33,864,294 −33,819,694 0.0000000000000% −33,819,693−33,775,093 0.0000000000000% −33,775,091 −33,730,491 0.0000000000000%−33,730,489 −33,685,889 0.0000000000000% −33,685,888 −33,641,2880.0000000000000% −33,641,286 −33,596,686 0.0000000000000% −33,596,684−33,552,084 0.0000000000000% −33,552,083 −33,507,483 0.0000000000000%−33,507,481 −33,462,881 0.0000000000000% −33,462,879 −33,418,2790.0000000000000% −33,418,278 −33,373,678 0.0000000000000% −33,373,676−33,329,076 0.0000000000000% −33,329,074 −33,284,474 0.0000000000000%−33,284,473 −33,239,873 0.0000000000000% −33,239,871 −33,195,2710.0000000000000% −33,195,269 −33,150,669 0.0000000000000% −33,150,668−33,106,068 0.0000000000000% −33,106,066 −33,061,466 0.0000000000000%−33,061,464 −33,016,864 0.0000000000000% −33,016,863 −32,972,2630.0000000000000% −32,972,261 −32,927,661 0.0000000000000% −32,927,659−32,883,059 0.0000000000000% −32,883,057 −32,838,457 0.0000000000000%−32,838,456 −32,793,856 0.0000000000000% −32,793,854 −32,749,2540.0000000000000% −32,749,252 −32,704,652 0.0000000000000% −32,704,651−32,660,051 0.0000000000000% −32,660,049 −32,615,449 0.0000000000000%−32,615,447 −32,570,847 0.0000000000000% −32,570,846 −32,526,2460.0000000000000% −32,526,244 −32,481,644 0.0000000000000% −32,481,642−32,437,042 0.0000000000000% −32,437,041 −32,392,441 0.0000000000000%−32,392,439 −32,347,839 0.0000000000000% −32,347,837 −32,303,2370.0000000000000% −32,303,236 −32,258,636 0.0000000000000% −32,258,634−32,214,034 0.0000000000000% −32,214,032 −32,169,432 0.0000000000000%−32,169,431 −32,124,831 0.0000000000000% −32,124,829 −32,080,2290.0000000000000% −32,080,227 −32,035,627 0.0000000000000% −32,035,626−31,991,026 0.0000000000000% −31,991,024 −31,946,424 0.0000000000000%−31,946,422 −31,901,822 0.0000000000000% −31,901,821 −31,857,2210.0000000000000% −31,857,219 −31,812,619 0.0000000000000% −31,812,617−31,768,017 0.0000000000000% −31,768,016 −31,723,416 0.0000000000000%−31,723,414 −31,678,814 0.0000000000000% −31,678,812 −31,634,2120.0000000000000% −31,634,211 −31,589,611 0.0000000000000% −31,589,609−31,545,009 0.0000000000000% −31,545,007 −31,500,407 0.0000000000000%−31,500,406 −31,455,806 0.0000000000000% −31,455,804 −31,411,2040.0000000000000% −31,411,202 −31,366,602 0.0000000000000% −31,366,601−31,322,001 0.0000000000000% −31,321,999 −31,277,399 0.0000000000000%−31,277,397 −31,232,797 0.0000000000000% −31,232,796 −31,188,1960.0000000000000% −31,188,194 −31,143,594 0.0000000000000% −31,143,592−31,098,992 0.0000000000000% −31,098,991 −31,054,391 0.0000000000000%−31,054,389 −31,009,789 0.0000000000000% −31,009,787 −30,965,1870.0000000000000% −30,965,186 −30,920,586 0.0000000000000% −30,920,584−30,875,984 0.0000000000000% −30,875,982 −30,831,382 0.0000000000000%−30,831,380 −30,786,780 0.0000000000000% −30,786,779 −30,742,1790.0000000000000% −30,742,177 −30,697,577 0.0000000000000% −30,697,575−30,652,975 0.0000000000000% −30,652,974 −30,608,374 0.0000000000000%−30,608,372 −30,563,772 0.0000000000000% −30,563,770 −30,519,1700.0000000000000% −30,519,169 −30,474,569 0.0000000000000% −30,474,567−30,429,967 0.0000000000000% −30,429,965 −30,385,365 0.0000000000000%−30,385,364 −30,340,764 0.0000000000000% −30,340,762 −30,296,1620.0000000000000% −30,296,160 −30,251,560 0.0000000000000% −30,251,559−30,206,959 0.0000000000000% −30,206,957 −30,162,357 0.0000000000000%−30,162,355 −30,117,755 0.0000000000000% −30,117,754 −30,073,1540.0000000000000% −30,073,152 −30,028,552 0.0000000000000% −30,028,550−29,983,950 0.0000000000000% −29,983,949 −29,939,349 0.0000000000000%−29,939,347 −29,894,747 0.0000000000000% −29,894,745 −29,850,1450.0000000000000% −29,850,144 −29,805,544 0.0000000000000% −29,805,542−29,760,942 0.0000000000000% −29,760,940 −29,716,340 0.0000000000000%−29,716,339 −29,671,739 0.0000000000000% −29,671,737 −29,627,1370.0000000000000% −29,627,135 −29,582,535 0.0000000000000% −29,582,534−29,537,934 0.0000000000000% −29,537,932 −29,493,332 0.0000000000000%−29,493,330 −29,448,730 0.0000000000000% −29,448,729 −29,404,1290.0000000000000% −29,404,127 −29,359,527 0.0000000000000% −29,359,525−29,314,925 0.0000000000000% −29,314,924 −29,270,324 0.0000000000000%−29,270,322 −29,225,722 0.0000000000000% −29,225,720 −29,181,1200.0000000000000% −29,181,119 −29,136,519 0.0000000000000% −29,136,517−29,091,917 0.0000000000000% −29,091,915 −29,047,315 0.0000000000000%−29,047,314 −29,002,714 0.0000000000000% −29,002,712 −28,958,1120.0000000000000% −28,958,110 −28,913,510 0.0000000000000% −28,913,509−28,868,909 0.0000000000000% −28,868,907 −28,824,307 0.0000000000000%−28,824,305 −28,779,705 0.0000000000000% −28,779,703 −28,735,1030.0000000000000% −28,735,102 −28,690,502 0.0000000000000% −28,690,500−28,645,900 0.0000000000000% 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0.0000000000000% −12,187,881−12,143,281 0.0000000000000% −12,143,279 −12,098,679 0.0000000000000%−12,098,677 −12,054,077 0.0000000000000% −12,054,076 −12,009,4760.0000000000000% −12,009,474 −11,964,874 0.0000000000000% −11,964,872−11,920,272 0.0000000000000% −11,920,271 −11,875,671 0.0000000000000%−11,875,669 −11,831,069 0.0000000000000% −11,831,067 −11,786,4670.0000000000000% −11,786,466 −11,741,866 0.0000000000000% −11,741,864−11,697,264 0.0000000000000% −11,697,262 −11,652,662 0.0000000000000%−11,652,661 −11,608,061 0.0000000000000% −11,608,059 −11,563,4590.0000000000000% −11,563,457 −11,518,857 0.0000000000000% −11,518,856−11,474,256 0.0000000000000% −11,474,254 −11,429,654 0.0000000000000%−11,429,652 −11,385,052 0.0000000000000% −11,385,051 −11,340,4510.0000000000000% −11,340,449 −11,295,849 0.0000000000000% −11,295,847−11,251,247 0.0000000000000% −11,251,246 −11,206,646 0.0000000000000%−11,206,644 −11,162,044 0.0000000000000% −11,162,042 −11,117,4420.0000000000000% −11,117,441 −11,072,841 0.0000000000000% −11,072,839−11,028,239 0.0000000000000% −11,028,237 −10,983,637 0.0000000000000%−10,983,636 −10,939,036 0.0000000000000% −10,939,034 −10,894,4340.0000000000000% −10,894,432 −10,849,832 0.0000000000000% −10,849,831−10,805,231 0.0000000000000% −10,805,229 −10,760,629 0.0000000000000%−10,760,627 −10,716,027 0.0000000000000% −10,716,026 −10,671,4260.0000000000000% −10,671,424 −10,626,824 0.0000000000000% −10,626,822−10,582,222 0.0000000000000% −10,582,221 −10,537,621 0.0000000000000%−10,537,619 −10,493,019 0.0000000000000% −10,493,017 −10,448,4170.0000000000000% −10,448,415 −10,403,815 0.0000000000000% −10,403,814−10,359,214 0.0000000000000% −10,359,212 −10,314,612 0.0000000000000%−10,314,610 −10,270,010 0.0000000000000% −10,270,009 −10,225,4090.0000000000000% −10,225,407 −10,180,807 0.0000000000000% −10,180,805−10,136,205 0.0000000000000% −10,136,204 −10,091,604 0.0000000000000%−10,091,602 −10,047,002 0.0000000000000% −10,047,000 −10,002,4000.0000000000000% −10,002,399 −9,957,799 0.0000000000000% −9,957,797−9,913,197 0.0000000000000% −9,913,195 −9,868,595 0.0000000000000%−9,868,594 −9,823,994 0.0000000000000% −9,823,992 −9,779,3920.0000000000000% −9,779,390 −9,734,790 0.0000000000000% −9,734,789−9,690,189 0.0000000000000% −9,690,187 −9,645,587 0.0000000000000%−9,645,585 −9,600,985 0.0000000000000% −9,600,984 −9,556,3840.0000000000000% −9,556,382 −9,511,782 0.0000000000000% −9,511,780−9,467,180 0.0000000000000% −9,467,179 −9,422,579 0.0000000000000%−9,422,577 −9,377,977 0.0000000000000% −9,377,975 −9,333,3750.0000000000000% −9,333,374 −9,288,774 0.0000000000000% −9,288,772−9,244,172 0.0000000000000% −9,244,170 −9,199,570 0.0000000000000%−9,199,569 −9,154,969 0.0000000000000% −9,154,967 −9,110,3670.0000000000000% −9,110,365 −9,065,765 0.0000000000000% −9,065,764−9,021,164 0.0000000000000% −9,021,162 −8,976,562 0.0000000000000%−8,976,560 −8,931,960 0.0000000000000% −8,931,959 −8,887,3590.0000000000000% −8,887,357 −8,842,757 0.0000000000000% −8,842,755−8,798,155 0.0000000000000% −8,798,154 −8,753,554 0.0000000000000%−8,753,552 −8,708,952 0.0000000000000% −8,708,950 −8,664,3500.0000000000000% −8,664,349 −8,619,749 0.0000000000000% −8,619,747−8,575,147 0.0000000000000% −8,575,145 −8,530,545 0.0000000000000%−8,530,544 −8,485,944 0.0000000000000% −8,485,942 −8,441,3420.0000000000000% −8,441,340 −8,396,740 0.0000000000000% −8,396,738−8,352,138 0.0000000000000% −8,352,137 −8,307,537 0.0000000000000%−8,307,535 −8,262,935 0.0000000000000% −8,262,933 −8,218,3330.0000000000000% −8,218,332 −8,173,732 0.0000000000000% −8,173,730−8,129,130 0.0000000000000% −8,129,128 −8,084,528 0.0000000000000%−8,084,527 −8,039,927 0.0000000000000% −8,039,925 −7,995,3250.0000000000000% −7,995,323 −7,950,723 0.0000000000000% −7,950,722−7,906,122 0.0000000000000% −7,906,120 −7,861,520 0.0000000000000%−7,861,518 −7,816,918 0.0000000000000% −7,816,917 −7,772,3170.0000000000000% −7,772,315 −7,727,715 0.0000000000000% −7,727,713−7,683,113 0.0000000000000% −7,683,112 −7,638,512 0.0000000000000%−7,638,510 −7,593,910 0.0000000000000% −7,593,908 −7,549,3080.0000000000000% −7,549,307 −7,504,707 0.0000000000000% −7,504,705−7,460,105 0.0000000000000% −7,460,103 −7,415,503 0.0000000000000%−7,415,502 −7,370,902 0.0000000000000% −7,370,900 −7,326,3000.0000000000000% −7,326,298 −7,281,698 0.0000000000000% −7,281,697−7,237,097 0.0000000000000% −7,237,095 −7,192,495 0.0000000000000%−7,192,493 −7,147,893 0.0000000000000% −7,147,892 −7,103,2920.0000000000000% −7,103,290 −7,058,690 0.0000000000000% −7,058,688−7,014,088 0.0000000000000% −7,014,087 −6,969,487 0.0000000000000%−6,969,485 −6,924,885 0.0000000000000% −6,924,883 −6,880,2830.0000000000000% −6,880,282 −6,835,682 0.0000000000000% −6,835,680−6,791,080 0.0000000000000% −6,791,078 −6,746,478 0.0000000000000%−6,746,477 −6,701,877 0.0000000000000% −6,701,875 −6,657,2750.0000000000000% −6,657,273 −6,612,673 0.0000000000000% −6,612,672−6,568,072 0.0000000000000% −6,568,070 −6,523,470 0.0000000000000%−6,523,468 −6,478,868 0.0000000000000% −6,478,867 −6,434,2670.0000000000000% −6,434,265 −6,389,665 0.0000000000000% −6,389,663−6,345,063 0.0000000000000% −6,345,061 −6,300,461 0.0000000000000%−6,300,460 −6,255,860 0.0000000000000% −6,255,858 −6,211,2580.0000000000000% −6,211,256 −6,166,656 0.0000000000000% −6,166,655−6,122,055 0.0000000000000% −6,122,053 −6,077,453 0.0000000000000%−6,077,451 −6,032,851 0.0000000000000% −6,032,850 −5,988,2500.0000000000000% −5,988,248 −5,943,648 0.0000000000000% −5,943,646−5,899,046 0.0000000000000% −5,899,045 −5,854,445 0.0000000000000%−5,854,443 −5,809,843 0.0000000000000% −5,809,841 −5,765,2410.0000000000000% −5,765,240 −5,720,640 0.0000000000000% −5,720,638−5,676,038 0.0000000000000% −5,676,036 −5,631,436 0.0000000000000%−5,631,435 −5,586,835 0.0000000000000% −5,586,833 −5,542,2330.0000000000000% −5,542,231 −5,497,631 0.0000000000000% −5,497,630−5,453,030 0.0000000000000% −5,453,028 −5,408,428 0.0000000000000%−5,408,426 −5,363,826 0.0000000000000% −5,363,825 −5,319,2250.0000000000000% −5,319,223 −5,274,623 0.0000000000000% −5,274,621−5,230,021 0.0000000000000% −5,230,020 −5,185,420 0.0000000000000%−5,185,418 −5,140,818 0.0000000000000% −5,140,816 −5,096,2160.0000000000000% −5,096,215 −5,051,615 0.0000000000000% −5,051,613−5,007,013 0.0000000000000% −5,007,011 −4,962,411 0.0000000000000%−4,962,410 −4,917,810 0.0000000000000% −4,917,808 −4,873,2080.0000000000000% −4,873,206 −4,828,606 0.0000000000000% −4,828,605−4,784,005 0.0000000000000% −4,784,003 −4,739,403 0.0000000000000%−4,739,401 −4,694,801 0.0000000000000% −4,694,800 −4,650,2000.0000000000000% −4,650,198 −4,605,598 0.0000000000000% −4,605,596−4,560,996 0.0000000000000% −4,560,995 −4,516,395 0.0000000000000%−4,516,393 −4,471,793 0.0000000000000% −4,471,791 −4,427,1910.0000000000000% −4,427,190 −4,382,590 0.0000000000000% −4,382,588−4,337,988 0.0000000000000% −4,337,986 −4,293,386 0.0000000000000%−4,293,384 −4,248,784 0.0000000000000% −4,248,783 −4,204,1830.0000000000000% −4,204,181 −4,159,581 0.0000000000000% −4,159,579−4,114,979 0.0000000000000% −4,114,978 −4,070,378 0.0000000000000%−4,070,376 −4,025,776 0.0000000000000% −4,025,774 −3,981,1740.0000000000000% −3,981,173 −3,936,573 0.0000000000000% −3,936,571−3,891,971 0.0000000000000% −3,891,969 −3,847,369 0.0000000000000%−3,847,368 −3,802,768 0.0000000000000% −3,802,766 −3,758,1660.0000000000000% −3,758,164 −3,713,564 0.0000000000000% −3,713,563−3,668,963 0.0000000000000% −3,668,961 −3,624,361 0.0000000000000%−3,624,359 −3,579,759 0.0000000000000% −3,579,758 −3,535,1580.0000000000000% −3,535,156 −3,490,556 0.0000000000000% −3,490,554−3,445,954 0.0000000000000% −3,445,953 −3,401,353 0.0000000000000%−3,401,351 −3,356,751 0.0000000000000% −3,356,749 −3,312,1490.0000000000000% −3,312,148 −3,267,548 0.0000000000000% −3,267,546−3,222,946 0.0000000000000% −3,222,944 −3,178,344 0.0000000000000%−3,178,343 −3,133,743 0.0000000000000% −3,133,741 −3,089,1410.0000000000000% −3,089,139 −3,044,539 0.0000000000000% −3,044,538−2,999,938 0.0000000000000% −2,999,936 −2,955,336 0.0000000000000%−2,955,334 −2,910,734 0.0000000000000% −2,910,733 −2,866,1330.0000000000000% −2,866,131 −2,821,531 0.0000000000000% −2,821,529−2,776,929 0.0000000000002% −2,776,928 −2,732,328 0.0000000000018%−2,732,326 −2,687,726 0.0000000000065% −2,687,724 −2,643,1240.0000000000219% −2,643,123 −2,598,523 0.0000000001017% −2,598,521−2,553,921 0.0000000003655% −2,553,919 −2,509,319 0.0000000009832%−2,509,318 −2,464,718 0.0000000027553% −2,464,716 −2,420,1160.0000000094423% −2,420,114 −2,375,514 0.0000000255127% −2,375,513−2,330,913 0.0000000583709% −2,330,911 −2,286,311 0.0000002203282%−2,286,309 −2,241,709 0.0000006592008% −2,241,707 −2,197,1070.0000028514222% −2,197,106 −2,152,506 0.0000049230940% −2,152,504−2,107,904 0.0000083704808% −2,107,902 −2,063,302 0.0000147843515%−2,063,301 −2,018,701 0.0000260338175% −2,018,699 −1,974,0990.0000433902708% −1,974,097 −1,929,497 0.0000751494837% −1,929,496−1,884,896 0.0001174352047% −1,884,894 −1,840,294 0.0002107401227%−1,840,292 −1,795,692 0.0003543454910% −1,795,691 −1,751,0910.0005322695270% −1,751,089 −1,706,489 0.0007929034140% −1,706,487−1,661,887 0.0011260084372% −1,661,886 −1,617,286 0.0016093991532%−1,617,284 −1,572,684 0.0022344726019% −1,572,682 −1,528,0820.0038838970629% −1,528,081 −1,483,481 0.0055739683663% −1,483,479−1,438,879 0.0073772863670% −1,438,877 −1,394,277 0.0098943650295%−1,394,276 −1,349,676 0.0143006184707% −1,349,674 −1,305,0740.0183460785112% −1,305,072 −1,260,472 0.0261547277456% −1,260,471−1,215,871 0.0326197573591% −1,215,869 −1,171,269 0.0425953458723%−1,171,267 −1,126,667 0.0532791541028% −1,126,666 −1,082,0660.0641757265385% −1,082,064 −1,037,464 0.0813017293983% −1,037,462−992,862 0.1109959473059% −992,861 −948,261 0.1458629976310% −948,259−903,659 0.1993542936404% −903,657 −859,057 0.2853471413593% −859,056−814,456 0.3477874325450% −814,454 −769,854 0.4252378540761% −769,852−725,252 0.5003644927948% −725,251 −680,651 0.6504013874467% −680,649−636,049 0.7638748188683% −636,047 −591,447 0.9285882632017% −591,446−546,846 1.1098746419447% −546,844 −502,244 1.2875854315004% −502,242−457,642 1.6835910404878% −457,641 −413,041 1.9565302556069% −413,039−368,439 2.3984238127086% −368,437 −323,837 2.9514412946921% −323,836−279,236 3.4425865251655% −279,234 −234,634 4.3555633509304% −234,632−190,032 5.0114075306107% −190,030 −145,430 5.6315527250100% −145,429−100,829 6.8636834340708% −100,827 −56,227 7.7953930073120% −56,225−11,625 9.6233601880087% −11,624 32,976 10.7166168500693% 32,978 77,57811.9991742114335% 77,580 122,180 13.5405065877200% 122,181 166,78114.8397231105370% 166,783 211,383 17.2710387405954% 211,385 255,98519.1142832254999% 255,986 300,586 21.6440319572670% 300,588 345,18823.6419140520004% 345,190 389,790 26.1660237764730% 389,791 434,39131.6868857458459% 434,393 478,993 35.4711252066020% 478,995 523,59538.5130187980694% 523,596 568,196 42.0279682320189% 568,198 612,79844.8696515698134% 612,800 657,400 47.4674173810816% 657,401 702,00150.3974553776583% 702,003 746,603 53.0472292653786% 746,605 791,20556.4432553966435% 791,206 835,806 61.3430697312952% 835,808 880,40864.2048100657864% 880,410 925,010 68.8752103906634% 925,011 969,61171.6614636000836% 969,613 1,014,213 74.1731669085254% 1,014,2151,058,815 80.1002653117845% 1,058,816 1,103,416 82.1420889430752%1,103,418 1,148,018 83.9004417117022% 1,148,020 1,192,62085.8838267577670% 1,192,621 1,237,221 87.2367841677128% 1,237,2231,281,823 88.8625917670604% 1,281,825 1,326,425 90.3051654466377%1,326,426 1,371,026 92.9204530812783% 1,371,028 1,415,62893.9405468081542% 1,415,630 1,460,230 94.8737949872860% 1,460,2311,504,831 95.6977286713466% 1,504,833 1,549,433 96.3011026856639%1,549,435 1,594,035 96.9124001310548% 1,594,036 1,638,63697.4821645032653% 1,638,638 1,683,238 97.9746466630464% 1,683,2401,727,840 98.7128389094456% 1,727,842 1,772,442 98.9803293287885%1,772,443 1,817,043 99.2617630786270% 1,817,045 1,861,64599.4130412604105% 1,861,647 1,906,247 99.5296821736390% 1,906,2481,950,848 99.6899419150264% 1,950,850 1,995,450 99.7560034168711%1,995,452 2,040,052 99.8149077928228% 2,040,053 2,084,65399.8578823240910% 2,084,655 2,129,255 99.8897335086087% 2,129,2572,173,857 99.9146184071978% 2,173,858 2,218,458 99.9387428287191%2,218,460 2,263,060 99.9591714611863% 2,263,062 2,307,66299.9784510536780% 2,307,663 2,352,263 99.9856071524159% 2,352,2652,396,865 99.9917753707933% 2,396,867 2,441,467 99.9941766602103%2,441,468 2,486,068 99.9958998396849% 2,486,070 2,530,67099.9972771437751% 2,530,672 2,575,272 99.9984679938238% 2,575,2732,619,873 99.9989904622893% 2,619,875 2,664,475 99.9993766063641%2,664,477 2,709,077 99.9995756749421% 2,709,078 2,753,67899.9997905951318% 2,753,680 2,798,280 99.9998693701572% 2,798,2822,842,882 99.9999342176618% 2,842,883 2,887,483 99.9999677509958%2,887,485 2,932,085 99.9999817757014% 2,932,087 2,976,68799.9999893038224% 2,976,688 3,021,288 99.9999957381375% 3,021,2903,065,890 99.9999977033306% 3,065,892 3,110,492 99.9999989187910%3,110,493 3,155,093 99.9999996001808% 3,155,095 3,199,69599.9999998600271% 3,199,697 3,244,297 99.9999999290552% 3,244,2983,288,898 99.9999999643476% 3,288,900 3,333,500 99.9999999844579%3,333,502 3,378,102 99.9999999936512% 3,378,103 3,422,70399.9999999972243% 3,422,705 3,467,305 99.9999999993124% 3,467,3073,511,907 99.9999999996973% 3,511,908 3,556,508 99.9999999999067%3,556,510 3,601,110 99.9999999999658% 3,601,112 3,645,71299.9999999999881% 3,645,713 3,690,313 99.9999999999996% 3,690,3153,734,915 100.0000000000000% 3,734,917 3,779,517 100.0000000000000%3,779,519 3,824,119 100.0000000000000% 3,824,120 3,868,720100.0000000000000% 3,868,722 3,913,322 100.0000000000000% 3,913,3243,957,924 100.0000000000000% 3,957,925 4,002,525 100.0000000000000%

Sample Data Set B Table of Outcome Intervals Outcome Intervals OutcomesIntervals From To As % Of All Outcomes −6,189,219 −6,180,3910.0000000000000% −6,180,391 −6,171,563 0.0000000000000% −6,171,563−6,162,735 0.0000000000000% −6,162,736 −6,153,908 0.0000000000000%−6,153,908 −6,145,080 0.0000000000000% −6,145,080 −6,136,2520.0000000000000% −6,136,252 −6,127,424 0.0000000000000% −6,127,424−6,118,596 0.0000000000000% −6,118,597 −6,109,769 0.0000000000000%−6,109,769 −6,100,941 0.0000000000000% −6,100,941 −6,092,1130.0000000000000% −6,092,113 −6,083,285 0.0000000000000% −6,083,285−6,074,457 0.0000000000000% −6,074,458 −6,065,630 0.0000000000000%−6,065,630 −6,056,802 0.0000000000000% −6,056,802 −6,047,9740.0000000000000% −6,047,974 −6,039,146 0.0000000000000% −6,039,146−6,030,318 0.0000000000000% −6,030,319 −6,021,491 0.0000000000000%−6,021,491 −6,012,663 0.0000000000000% −6,012,663 −6,003,8350.0000000000000% −6,003,835 −5,995,007 0.0000000000000% −5,995,007−5,986,179 0.0000000000000% −5,986,180 −5,977,352 0.0000000000000%−5,977,352 −5,968,524 0.0000000000000% −5,968,524 −5,959,6960.0000000000000% −5,959,696 −5,950,868 0.0000000000000% −5,950,868−5,942,040 0.0000000000000% −5,942,041 −5,933,213 0.0000000000000%−5,933,213 −5,924,385 0.0000000000000% −5,924,385 −5,915,5570.0000000000000% −5,915,557 −5,906,729 0.0000000000000% −5,906,730−5,897,902 0.0000000000000% −5,897,902 −5,889,074 0.0000000000000%−5,889,074 −5,880,246 0.0000000000000% −5,880,246 −5,871,4180.0000000000000% −5,871,418 −5,862,590 0.0000000000000% −5,862,591−5,853,763 0.0000000000000% −5,853,763 −5,844,935 0.0000000000000%−5,844,935 −5,836,107 0.0000000000000% −5,836,107 −5,827,2790.0000000000000% −5,827,279 −5,818,451 0.0000000000000% −5,818,452−5,809,624 0.0000000000000% −5,809,624 −5,800,796 0.0000000000000%−5,800,796 −5,791,968 0.0000000000000% −5,791,968 −5,783,1400.0000000000000% −5,783,140 −5,774,312 0.0000000000000% −5,774,313−5,765,485 0.0000000000000% −5,765,485 −5,756,657 0.0000000000000%−5,756,657 −5,747,829 0.0000000000000% −5,747,829 −5,739,0010.0000000000000% −5,739,001 −5,730,173 0.0000000000000% −5,730,174−5,721,346 0.0000000000000% −5,721,346 −5,712,518 0.0000000000000%−5,712,518 −5,703,690 0.0000000000000% −5,703,690 −5,694,8620.0000000000000% −5,694,862 −5,686,034 0.0000000000000% −5,686,035−5,677,207 0.0000000000000% −5,677,207 −5,668,379 0.0000000000000%−5,668,379 −5,659,551 0.0000000000000% −5,659,551 −5,650,7230.0000000000000% −5,650,723 −5,641,895 0.0000000000000% −5,641,896−5,633,068 0.0000000000000% −5,633,068 −5,624,240 0.0000000000000%−5,624,240 −5,615,412 0.0000000000000% −5,615,412 −5,606,5840.0000000000000% −5,606,584 −5,597,756 0.0000000000000% −5,597,757−5,588,929 0.0000000000000% −5,588,929 −5,580,101 0.0000000000000%−5,580,101 −5,571,273 0.0000000000000% −5,571,273 −5,562,4450.0000000000000% −5,562,445 −5,553,617 0.0000000000000% −5,553,618−5,544,790 0.0000000000000% −5,544,790 −5,535,962 0.0000000000000%−5,535,962 −5,527,134 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78,5184.4317192511575% 78,517 87,345 4.8620295708883% 87,345 96,1735.3227757643496% 96,173 105,001 5.9201079178989% 105,001 113,8297.0200666288558% 113,829 122,657 7.6771709610899% 122,656 131,4848.5197218917224% 131,484 140,312 9.2739644382429% 140,312 149,14010.4220567986609% 149,140 157,968 11.2026975791171% 157,968 166,79611.9636698324646% 166,795 175,623 12.8994221640359% 175,623 184,45114.5591526657690% 184,451 193,279 15.7527468505712% 193,279 202,10716.7066541951429% 202,107 210,935 17.8352104858872% 210,934 219,76219.3619355129900% 219,762 228,590 20.4719997581447% 228,590 237,41821.5752574788502% 237,418 246,246 22.7616879111725% 246,246 255,07425.0250574227471% 255,073 263,901 26.1738774322370% 263,901 272,72927.3342108574833% 272,729 281,557 28.4974928602907% 281,557 290,38529.9710838300900% 290,385 299,213 31.2715008713569% 299,212 308,04032.5624227981239% 308,040 316,868 34.3661080347948% 316,868 325,69637.1901192430042% 325,696 334,524 38.6944177806645% 334,524 343,35241.1261490772473% 343,351 352,179 42.4970517110926% 352,179 361,00743.8439742004818% 361,007 369,835 45.7646119773856% 369,835 378,66347.1170087038107% 378,663 387,491 48.6189759297485% 387,490 396,31850.4074353851961% 396,318 405,146 53.0631749379326% 405,146 413,97454.4501025847708% 413,974 422,802 55.7196116070101% 422,802 431,63056.9861500234953% 431,629 440,457 58.3248083103937% 440,457 449,28559.5598368417797% 449,285 458,113 60.8313804154635% 458,113 466,94162.4692034262420% 466,941 475,769 65.1336992940237% 475,768 484,59666.8047945442478% 484,596 493,424 67.9116366812887% 493,424 502,25269.0547528228996% 502,252 511,080 70.2423967415646% 511,080 519,90871.2800770724535% 519,907 528,735 72.2940241863967% 528,735 537,56373.2808090297395% 537,563 546,391 76.1210300178199% 546,391 555,21977.1619840597711% 555,219 564,047 78.7298841516823% 564,046 572,87479.7025751752425% 572,874 581,702 80.5962971065439% 581,702 590,53081.4312257181555% 590,530 599,358 82.2876181186992% 599,358 608,18683.0465289766277% 608,185 617,013 83.7883879998699% 617,013 625,84185.2008070244174% 625,841 634,669 86.0267757062123% 634,669 643,49786.6344298631061% 643,497 652,325 87.2962831344475% 652,324 661,15288.1100871645139% 661,152 669,980 88.6744814390339% 669,980 678,80889.2329968395081% 678,808 687,636 89.8370818689095% 687,636 696,46490.8264404017443% 696,463 705,291 91.3004754060173% 705,291 714,11991.7568123881555% 714,119 722,947 92.1954993684822% 722,947 731,77592.7085439720014% 731,775 740,603 93.5009567022676% 740,602 749,43093.8972918046852% 749,430 758,258 94.2251917390744% 758,258 767,08694.9135684384200% 767,086 775,914 95.1851783783244% 775,914 784,74295.4400832024829% 784,741 793,569 95.6855612340689% 793,569 802,39795.9240463782977% 802,397 811,225 96.2393422447117% 811,225 820,05396.4667929615409% 820,053 828,881 96.7015694209675% 828,880 837,70897.0184083126487% 837,708 846,536 97.3446509419694% 846,536 855,36497.5172975593502% 855,364 864,192 97.6666441867764% 864,191 873,01997.8098514235311% 873,019 881,847 97.9422637945395% 881,847 890,67598.0665222997669% 890,675 899,503 98.1812497688883% 899,503 908,33198.3177744072223% 908,330 917,158 98.5366951926678% 917,158 925,98698.6716389616476% 925,986 934,814 98.8295770262078% 934,814 943,64298.9153351587665% 943,642 952,470 98.9904929019223% 952,469 961,29799.0745476915369% 961,297 970,125 99.1373768010056% 970,125 978,95399.1962422068002% 978,953 987,781 99.3150171289429% 987,781 996,60999.3659767489186% 996,608 1,005,436 99.4237261195182% 1,005,4361,014,264 99.4656510384189% 1,014,264 1,023,092 99.5032386574159%1,023,092 1,031,920 99.5380642769567% 1,031,920 1,040,74899.5725822463177% 1,040,747 1,049,575 99.6026384320762% 1,049,5751,058,403 99.6600704973477% 1,058,403 1,067,231 99.6863732068944%1,067,231 1,076,059 99.7176987469655% 1,076,059 1,084,88799.7392007669498% 1,084,886 1,093,714 99.7611893510610% 1,093,7141,102,542 99.7867158364390% 1,102,542 1,111,370 99.8026403958946%1,111,370 1,120,198 99.8182945822592% 1,120,198 1,129,02699.8335879704217% 1,129,025 1,137,853 99.8662893783230% 1,137,8531,146,681 99.8762876291834% 1,146,681 1,155,509 99.8876961075898%1,155,509 1,164,337 99.8986580532362% 1,164,337 1,173,16599.9084014493729% 1,173,164 1,181,992 99.9189121766062% 1,181,9921,190,820 99.9263113815090% 1,190,820 1,199,648 99.9325875922218%1,199,648 1,208,476 99.9440771598048% 1,208,476 1,217,30499.9488326313787% 1,217,303 1,226,131 99.9531501238360% 1,226,1311,234,959 99.9572135667362% 1,234,959 1,243,787 99.9607771221150%1,243,787 1,252,615 99.9648885992668% 1,252,615 1,261,44399.9677993372639% 1,261,442 1,270,270 99.9708277833101% 1,270,2701,279,098 99.9766768471747% 1,279,098 1,287,926 99.9788027574589%1,287,926 1,296,754 99.9809915975477% 1,296,754 1,305,58299.9827315011345% 1,305,581 1,314,409 99.9843591057442% 1,314,4091,323,237 99.9858822416518% 1,323,237 1,332,065 99.9882130049427%1,332,065 1,340,893 99.9895674869626% 1,340,893 1,349,72199.9906811254479% 1,349,720 1,358,548 99.9928632490326% 1,358,5481,367,376 99.9935505961091% 1,367,376 1,376,204 99.9941852572087%1,376,204 1,385,032 99.9947919034920% 1,385,032 1,393,86099.9952843268296% 1,393,859 1,402,687 99.9957372772911% 1,402,6871,411,515 99.9961548559623% 1,411,515 1,420,343 99.9965647646795%1,420,343 1,429,171 99.9973429142186% 1,429,171 1,437,99999.9976648553721% 1,437,998 1,446,826 99.9980023800821% 1,446,8261,455,654 99.9982262654528% 1,455,654 1,464,482 99.9984182208428%1,464,482 1,473,310 99.9986030275837% 1,473,310 1,482,13899.9987569518289% 1,482,137 1,490,965 99.9988902475917% 1,490,9651,499,793 99.9991126173635% 1,499,793 1,508,621 99.9992062669317%1,508,621 1,517,449 99.9993128170879% 1,517,449 1,526,27799.9994497585909% 1,526,276 1,535,104 99.9995363858439% 1,535,1041,543,932 99.9995966401987% 1,543,932 1,552,760 99.9996557201233%1,552,760 1,561,588 99.9997014806982% 1,561,588 1,570,41699.9997375844423% 1,570,415 1,579,243 99.9997978471211% 1,579,2431,588,071 99.9998219917365% 1,588,071 1,596,899 99.9998458196008%1,596,899 1,605,727 99.9998669063356% 1,605,727 1,614,55599.9998843414189% 1,614,554 1,623,382 99.9999033158913% 1,623,3821,632,210 99.9999159546760% 1,632,210 1,641,038 99.9999264783932%1,641,038 1,649,866 99.9999442205832% 1,649,866 1,658,69499.9999516213197% 1,658,693 1,667,521 99.9999579611379% 1,667,5211,676,349 99.9999637070581% 1,676,349 1,685,177 99.9999687437232%1,685,177 1,694,005 99.9999742397635% 1,694,005 1,702,83399.9999777439142% 1,702,832 1,711,660 99.9999809476015% 1,711,6601,720,488 99.9999882392160% 1,720,488 1,729,316 99.9999899706132%1,729,316 1,738,144 99.9999913478751% 1,738,143 1,746,97199.9999927239400% 1,746,971 1,755,799 99.9999936952880% 1,755,7991,764,627 99.9999945428090% 1,764,627 1,773,455 99.9999953218466%1,773,455 1,782,283 99.9999961692339% 1,782,282 1,791,11099.9999969541865% 1,791,110 1,799,938 99.9999978939964% 1,799,9381,808,766 99.9999982148805% 1,808,766 1,817,594 99.9999984975014%1,817,594 1,826,422 99.9999987493247% 1,826,421 1,835,24999.9999989407763% 1,835,249 1,844,077 99.9999991028866% 1,844,0771,852,905 99.9999992409604% 1,852,905 1,861,733 99.9999993560209%1,861,733 1,870,561 99.9999995616141% 1,870,560 1,879,38899.9999996375787% 1,879,388 1,888,216 99.9999997141931% 1,888,2161,897,044 99.9999997626810% 1,897,044 1,905,872 99.9999998031043%1,905,872 1,914,700 99.9999998388010% 1,914,699 1,923,52799.9999998854099% 1,923,527 1,932,355 99.9999999057941% 1,932,3551,941,183 99.9999999386142% 1,941,183 1,950,011 99.9999999520031%1,950,011 1,958,839 99.9999999607930% 1,958,838 1,967,66699.9999999682489% 1,967,666 1,976,494 99.9999999758635% 1,976,4941,985,322 99.9999999801048% 1,985,322 1,994,150 99.9999999836841%1,994,150 2,002,978 99.9999999867806% 2,002,977 2,011,80599.9999999910593% 2,011,805 2,020,633 99.9999999927231% 2,020,6332,029,461 99.9999999942139% 2,029,461 2,038,289 99.9999999956640%2,038,289 2,047,117 99.9999999965226% 2,047,116 2,055,94499.9999999972914% 2,055,944 2,064,772 99.9999999979954% 2,064,7722,073,600 99.9999999984245% 2,073,600 2,082,428 99.9999999987502%2,082,428 2,091,256 99.9999999992291% 2,091,255 2,100,08399.9999999993852% 2,100,083 2,108,911 99.9999999995092% 2,108,9112,117,739 99.9999999996716% 2,117,739 2,126,567 99.9999999997533%2,126,567 2,135,395 99.9999999998092% 2,135,394 2,144,22299.9999999998609% 2,144,222 2,153,050 99.9999999999035% 2,153,0502,161,878 99.9999999999439% 2,161,878 2,170,706 99.9999999999585%2,170,706 2,179,534 99.9999999999686% 2,179,533 2,188,36199.9999999999760% 2,188,361 2,197,189 99.9999999999816% 2,197,1892,206,017 99.9999999999859% 2,206,017 2,214,845 99.9999999999893%2,214,845 2,223,673 99.9999999999925% 2,223,672 2,232,50099.9999999999962% 2,232,500 2,241,328 99.9999999999972% 2,241,3282,250,156 99.9999999999979% 2,250,156 2,258,984 99.9999999999984%2,258,984 2,267,812 99.9999999999989% 2,267,811 2,276,63999.9999999999992% 2,276,639 2,285,467 99.9999999999994% 2,285,4672,294,295 99.9999999999996% 2,294,295 2,303,123 99.9999999999997%2,303,123 2,311,951 99.9999999999999% 2,311,950 2,320,778100.0000000000000% 2,320,778 2,329,606 100.0000000000000% 2,329,6062,338,434 100.0000000000000% 2,338,434 2,347,262 100.0000000000000%2,347,262 2,356,090 100.0000000000000% 2,356,089 2,364,917100.0000000000000% 2,364,917 2,373,745 100.0000000000000% 2,373,7452,382,573 100.0000000000000% 2,382,573 2,391,401 100.0000000000000%2,391,401 2,400,229 100.0000000000000% 2,400,228 2,409,056100.0000000000000% 2,409,056 2,417,884 100.0000000000000% 2,417,8842,426,712 100.0000000000000% 2,426,712 2,435,540 100.0000000000000%2,435,540 2,444,368 100.0000000000000% 2,444,367 2,453,195100.0000000000000% 2,453,195 2,462,023 100.0000000000000% 2,462,0232,470,851 100.0000000000000% 2,470,851 2,479,679 100.0000000000000%2,479,679 2,488,507 100.0000000000000% 2,488,506 2,497,334100.0000000000000% 2,497,334 2,506,162 100.0000000000000% 2,506,1622,514,990 100.0000000000000% 2,514,990 2,523,818 100.0000000000000%2,523,818 2,532,646 100.0000000000000% 2,532,645 2,541,473100.0000000000000% 2,541,473 2,550,301 100.0000000000000% 2,550,3012,559,129 100.0000000000000% 2,559,129 2,567,957 100.0000000000000%2,567,956 2,576,784 100.0000000000000% 2,576,784 2,585,612100.0000000000000% 2,585,612 2,594,440 100.0000000000000% 2,594,4402,603,268 100.0000000000000% 2,603,268 2,612,096 100.0000000000000%2,612,095 2,620,923 100.0000000000000% 2,620,923 2,629,751100.0000000000000% 2,629,751 2,638,579 100.0000000000000%

Sample Data Sets A & B Table of Convolution Distributions OutcomesOutcome Intervals Outcome Intervals From To As % of All Outcomes−46,788,367 −46,734,937 0.0000000000000% −46,734,938 −46,681,5080.0000000000000% −46,681,508 −46,628,078 0.0000000000000% −46,628,079−46,574,649 0.0000000000000% −46,574,649 −46,521,219 0.0000000000000%−46,521,220 −46,467,790 0.0000000000000% −46,467,790 −46,414,3600.0000000000000% −46,414,361 −46,360,931 0.0000000000000% −46,360,931−46,307,501 0.0000000000000% −46,307,502 −46,254,072 0.0000000000000%−46,254,072 −46,200,642 0.0000000000000% −46,200,643 −46,147,2130.0000000000000% −46,147,213 −46,093,783 0.0000000000000% −46,093,784−46,040,354 0.0000000000000% −46,040,355 −45,986,925 0.0000000000000%−45,986,925 −45,933,495 0.0000000000000% −45,933,496 −45,880,0660.0000000000000% −45,880,066 −45,826,636 0.0000000000000% −45,826,637−45,773,207 0.0000000000000% −45,773,207 −45,719,777 0.0000000000000%−45,719,778 −45,666,348 0.0000000000000% −45,666,348 −45,612,9180.0000000000000% −45,612,919 −45,559,489 0.0000000000000% −45,559,489−45,506,059 0.0000000000000% −45,506,060 −45,452,630 0.0000000000000%−45,452,630 −45,399,200 0.0000000000000% −45,399,201 −45,345,7710.0000000000000% −45,345,771 −45,292,341 0.0000000000000% −45,292,342−45,238,912 0.0000000000000% −45,238,912 −45,185,482 0.0000000000000%−45,185,483 −45,132,053 0.0000000000000% −45,132,053 −45,078,6230.0000000000000% −45,078,624 −45,025,194 0.0000000000000% −45,025,195−44,971,765 0.0000000000000% −44,971,765 −44,918,335 0.0000000000000%−44,918,336 −44,864,906 0.0000000000000% −44,864,906 −44,811,4760.0000000000000% −44,811,477 −44,758,047 0.0000000000000% −44,758,047−44,704,617 0.0000000000000% −44,704,618 −44,651,188 0.0000000000000%−44,651,188 −44,597,758 0.0000000000000% −44,597,759 −44,544,3290.0000000000000% −44,544,329 −44,490,899 0.0000000000000% −44,490,900−44,437,470 0.0000000000000% −44,437,470 −44,384,040 0.0000000000000%−44,384,041 −44,330,611 0.0000000000000% −44,330,611 −44,277,1810.0000000000000% −44,277,182 −44,223,752 0.0000000000000% −44,223,752−44,170,322 0.0000000000000% −44,170,323 −44,116,893 0.0000000000000%−44,116,894 −44,063,464 0.0000000000000% −44,063,464 −44,010,0340.0000000000000% −44,010,035 −43,956,605 0.0000000000000% −43,956,605−43,903,175 0.0000000000000% −43,903,176 −43,849,746 0.0000000000000%−43,849,746 −43,796,316 0.0000000000000% −43,796,317 −43,742,8870.0000000000000% −43,742,887 −43,689,457 0.0000000000000% −43,689,458−43,636,028 0.0000000000000% −43,636,028 −43,582,598 0.0000000000000%−43,582,599 −43,529,169 0.0000000000000% −43,529,169 −43,475,7390.0000000000000% −43,475,740 −43,422,310 0.0000000000000% −43,422,310−43,368,880 0.0000000000000% −43,368,881 −43,315,451 0.0000000000000%−43,315,451 −43,262,021 0.0000000000000% −43,262,022 −43,208,5920.0000000000000% −43,208,592 −43,155,162 0.0000000000000% −43,155,163−43,101,733 0.0000000000000% −43,101,734 −43,048,304 0.0000000000000%−43,048,304 −42,994,874 0.0000000000000% −42,994,875 −42,941,4450.0000000000000% −42,941,445 −42,888,015 0.0000000000000% −42,888,016−42,834,586 0.0000000000000% −42,834,586 −42,781,156 0.0000000000000%−42,781,157 −42,727,727 0.0000000000000% −42,727,727 −42,674,2970.0000000000000% −42,674,298 −42,620,868 0.0000000000000% −42,620,868−42,567,438 0.0000000000000% −42,567,439 −42,514,009 0.0000000000000%−42,514,009 −42,460,579 0.0000000000000% −42,460,580 −42,407,1500.0000000000000% −42,407,150 −42,353,720 0.0000000000000% −42,353,721−42,300,291 0.0000000000000% −42,300,291 −42,246,861 0.0000000000000%−42,246,862 −42,193,432 0.0000000000000% −42,193,433 −42,140,0030.0000000000000% −42,140,003 −42,086,573 0.0000000000000% −42,086,574−42,033,144 0.0000000000000% −42,033,144 −41,979,714 0.0000000000000%−41,979,715 −41,926,285 0.0000000000000% −41,926,285 −41,872,8550.0000000000000% −41,872,856 −41,819,426 0.0000000000000% −41,819,426−41,765,996 0.0000000000000% −41,765,997 −41,712,567 0.0000000000000%−41,712,567 −41,659,137 0.0000000000000% −41,659,138 −41,605,7080.0000000000000% −41,605,708 −41,552,278 0.0000000000000% −41,552,279−41,498,849 0.0000000000000% −41,498,849 −41,445,419 0.0000000000000%−41,445,420 −41,391,990 0.0000000000000% −41,391,990 −41,338,5600.0000000000000% −41,338,561 −41,285,131 0.0000000000000% −41,285,132−41,231,702 0.0000000000000% −41,231,702 −41,178,272 0.0000000000000%−41,178,273 −41,124,843 0.0000000000000% −41,124,843 −41,071,4130.0000000000000% −41,071,414 −41,017,984 0.0000000000000% −41,017,984−40,964,554 0.0000000000000% −40,964,555 −40,911,125 0.0000000000000%−40,911,125 −40,857,695 0.0000000000000% −40,857,696 −40,804,2660.0000000000000% −40,804,266 −40,750,836 0.0000000000000% −40,750,837−40,697,407 0.0000000000000% −40,697,407 −40,643,977 0.0000000000000%−40,643,978 −40,590,548 0.0000000000000% −40,590,548 −40,537,1180.0000000000000% −40,537,119 −40,483,689 0.0000000000000% −40,483,689−40,430,259 0.0000000000000% −40,430,260 −40,376,830 0.0000000000000%−40,376,830 −40,323,400 0.0000000000000% −40,323,401 −40,269,9710.0000000000000% −40,269,972 −40,216,542 0.0000000000000% −40,216,542−40,163,112 0.0000000000000% −40,163,113 −40,109,683 0.0000000000000%−40,109,683 −40,056,253 0.0000000000000% −40,056,254 −40,002,8240.0000000000000% −40,002,824 −39,949,394 0.0000000000000% −39,949,395−39,895,965 0.0000000000000% −39,895,965 −39,842,535 0.0000000000000%−39,842,536 −39,789,106 0.0000000000000% −39,789,106 −39,735,6760.0000000000000% −39,735,677 −39,682,247 0.0000000000000% −39,682,247−39,628,817 0.0000000000000% −39,628,818 −39,575,388 0.0000000000000%−39,575,388 −39,521,958 0.0000000000000% −39,521,959 −39,468,5290.0000000000000% −39,468,529 −39,415,099 0.0000000000000% −39,415,100−39,361,670 0.0000000000000% −39,361,671 −39,308,241 0.0000000000000%−39,308,241 −39,254,811 0.0000000000000% −39,254,812 −39,201,3820.0000000000000% −39,201,382 −39,147,952 0.0000000000000% −39,147,953−39,094,523 0.0000000000000% −39,094,523 −39,041,093 0.0000000000000%−39,041,094 −38,987,664 0.0000000000000% −38,987,664 −38,934,2340.0000000000000% −38,934,235 −38,880,805 0.0000000000000% −38,880,805−38,827,375 0.0000000000000% −38,827,376 −38,773,946 0.0000000000000%−38,773,946 −38,720,516 0.0000000000000% −38,720,517 −38,667,0870.0000000000000% −38,667,087 −38,613,657 0.0000000000000% −38,613,658−38,560,228 0.0000000000000% −38,560,228 −38,506,798 0.0000000000000%−38,506,799 −38,453,369 0.0000000000000% −38,453,370 −38,399,9400.0000000000000% −38,399,940 −38,346,510 0.0000000000000% −38,346,511−38,293,081 0.0000000000000% −38,293,081 −38,239,651 0.0000000000000%−38,239,652 −38,186,222 0.0000000000000% −38,186,222 −38,132,7920.0000000000000% −38,132,793 −38,079,363 0.0000000000000% −38,079,363−38,025,933 0.0000000000000% −38,025,934 −37,972,504 0.0000000000000%−37,972,504 −37,919,074 0.0000000000000% −37,919,075 −37,865,6450.0000000000000% −37,865,645 −37,812,215 0.0000000000000% −37,812,216−37,758,786 0.0000000000000% −37,758,786 −37,705,356 0.0000000000000%−37,705,357 −37,651,927 0.0000000000000% −37,651,927 −37,598,4970.0000000000000% −37,598,498 −37,545,068 0.0000000000000% −37,545,068−37,491,638 0.0000000000000% −37,491,639 −37,438,209 0.0000000000000%−37,438,210 −37,384,780 0.0000000000000% −37,384,780 −37,331,3500.0000000000000% −37,331,351 −37,277,921 0.0000000000000% −37,277,921−37,224,491 0.0000000000000% −37,224,492 −37,171,062 0.0000000000000%−37,171,062 −37,117,632 0.0000000000000% −37,117,633 −37,064,2030.0000000000000% −37,064,203 −37,010,773 0.0000000000000% −37,010,774−36,957,344 0.0000000000000% −36,957,344 −36,903,914 0.0000000000000%−36,903,915 −36,850,485 0.0000000000000% −36,850,485 −36,797,0550.0000000000000% −36,797,056 −36,743,626 0.0000000000000% −36,743,626−36,690,196 0.0000000000000% −36,690,197 −36,636,767 0.0000000000000%−36,636,767 −36,583,337 0.0000000000000% −36,583,338 −36,529,9080.0000000000000% −36,529,909 −36,476,479 0.0000000000000% −36,476,479−36,423,049 0.0000000000000% −36,423,050 −36,369,620 0.0000000000000%−36,369,620 −36,316,190 0.0000000000000% −36,316,191 −36,262,7610.0000000000000% −36,262,761 −36,209,331 0.0000000000000% −36,209,332−36,155,902 0.0000000000000% −36,155,902 −36,102,472 0.0000000000000%−36,102,473 −36,049,043 0.0000000000000% −36,049,043 −35,995,6130.0000000000000% −35,995,614 −35,942,184 0.0000000000000% −35,942,184−35,888,754 0.0000000000000% −35,888,755 −35,835,325 0.0000000000000%−35,835,325 −35,781,895 0.0000000000000% −35,781,896 −35,728,4660.0000000000000% −35,728,466 −35,675,036 0.0000000000000% −35,675,037−35,621,607 0.0000000000000% −35,621,607 −35,568,177 0.0000000000000%−35,568,178 −35,514,748 0.0000000000000% −35,514,749 −35,461,3190.0000000000000% −35,461,319 −35,407,889 0.0000000000000% −35,407,890−35,354,460 0.0000000000000% −35,354,460 −35,301,030 0.0000000000000%−35,301,031 −35,247,601 0.0000000000000% −35,247,601 −35,194,1710.0000000000000% −35,194,172 −35,140,742 0.0000000000000% −35,140,742−35,087,312 0.0000000000000% −35,087,313 −35,033,883 0.0000000000000%−35,033,883 −34,980,453 0.0000000000000% −34,980,454 −34,927,0240.0000000000000% −34,927,024 −34,873,594 0.0000000000000% −34,873,595−34,820,165 0.0000000000000% −34,820,165 −34,766,735 0.0000000000000%−34,766,736 −34,713,306 0.0000000000000% −34,713,306 −34,659,8760.0000000000000% −34,659,877 −34,606,447 0.0000000000000% −34,606,448−34,553,018 0.0000000000000% −34,553,018 −34,499,588 0.0000000000000%−34,499,589 −34,446,159 0.0000000000000% −34,446,159 −34,392,7290.0000000000000% −34,392,730 −34,339,300 0.0000000000000% −34,339,300−34,285,870 0.0000000000000% −34,285,871 −34,232,441 0.0000000000000%−34,232,441 −34,179,011 0.0000000000000% −34,179,012 −34,125,5820.0000000000000% −34,125,582 −34,072,152 0.0000000000000% −34,072,153−34,018,723 0.0000000000000% −34,018,723 −33,965,293 0.0000000000000%−33,965,294 −33,911,864 0.0000000000000% −33,911,864 −33,858,4340.0000000000000% −33,858,435 −33,805,005 0.0000000000000% −33,805,005−33,751,575 0.0000000000000% −33,751,576 −33,698,146 0.0000000000000%−33,698,147 −33,644,717 0.0000000000000% −33,644,717 −33,591,2870.0000000000000% −33,591,288 −33,537,858 0.0000000000000% −33,537,858−33,484,428 0.0000000000000% −33,484,429 −33,430,999 0.0000000000000%−33,430,999 −33,377,569 0.0000000000000% −33,377,570 −33,324,1400.0000000000000% −33,324,140 −33,270,710 0.0000000000000% −33,270,711−33,217,281 0.0000000000000% −33,217,281 −33,163,851 0.0000000000000%−33,163,852 −33,110,422 0.0000000000000% −33,110,422 −33,056,9920.0000000000000% −33,056,993 −33,003,563 0.0000000000000% −33,003,563−32,950,133 0.0000000000000% −32,950,134 −32,896,704 0.0000000000000%−32,896,704 −32,843,274 0.0000000000000% −32,843,275 −32,789,8450.0000000000000% −32,789,845 −32,736,415 0.0000000000000% −32,736,416−32,682,986 0.0000000000000% −32,682,987 −32,629,557 0.0000000000000%−32,629,557 −32,576,127 0.0000000000000% −32,576,128 −32,522,6980.0000000000000% −32,522,698 −32,469,268 0.0000000000000% −32,469,269−32,415,839 0.0000000000000% −32,415,839 −32,362,409 0.0000000000000%−32,362,410 −32,308,980 0.0000000000000% −32,308,980 −32,255,5500.0000000000000% −32,255,551 −32,202,121 0.0000000000000% −32,202,121−32,148,691 0.0000000000000% −32,148,692 −32,095,262 0.0000000000000%−32,095,262 −32,041,832 0.0000000000000% −32,041,833 −31,988,4030.0000000000000% −31,988,403 −31,934,973 0.0000000000000% −31,934,974−31,881,544 0.0000000000000% −31,881,544 −31,828,114 0.0000000000000%−31,828,115 −31,774,685 0.0000000000000% −31,774,686 −31,721,2560.0000000000000% −31,721,256 −31,667,826 0.0000000000000% −31,667,827−31,614,397 0.0000000000000% −31,614,397 −31,560,967 0.0000000000000%−31,560,968 −31,507,538 0.0000000000000% −31,507,538 −31,454,1080.0000000000000% −31,454,109 −31,400,679 0.0000000000000% −31,400,679−31,347,249 0.0000000000000% −31,347,250 −31,293,820 0.0000000000000%−31,293,820 −31,240,390 0.0000000000000% −31,240,391 −31,186,9610.0000000000000% −31,186,961 −31,133,531 0.0000000000000% −31,133,532−31,080,102 0.0000000000000% −31,080,102 −31,026,672 0.0000000000000%−31,026,673 −30,973,243 0.0000000000000% −30,973,243 −30,919,8130.0000000000000% −30,919,814 −30,866,384 0.0000000000000% −30,866,384−30,812,954 0.0000000000000% −30,812,955 −30,759,525 0.0000000000000%−30,759,526 −30,706,096 0.0000000000000% −30,706,096 −30,652,6660.0000000000000% −30,652,667 −30,599,237 0.0000000000000% −30,599,237−30,545,807 0.0000000000000% −30,545,808 −30,492,378 0.0000000000000%−30,492,378 −30,438,948 0.0000000000000% −30,438,949 −30,385,5190.0000000000000% −30,385,519 −30,332,089 0.0000000000000% −30,332,090−30,278,660 0.0000000000000% −30,278,660 −30,225,230 0.0000000000000%−30,225,231 −30,171,801 0.0000000000000% −30,171,801 −30,118,3710.0000000000000% −30,118,372 −30,064,942 0.0000000000000% −30,064,942−30,011,512 0.0000000000000% −30,011,513 −29,958,083 0.0000000000000%−29,958,083 −29,904,653 0.0000000000000% −29,904,654 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−6,288,827−6,235,397 0.0000000000000% −6,235,398 −6,181,968 0.0000000000000%−6,181,968 −6,128,538 0.0000000000000% −6,128,539 −6,075,1090.0000000000000% −6,075,110 −6,021,680 0.0000000000000% −6,021,680−5,968,250 0.0000000000000% −5,968,251 −5,914,821 0.0000000000000%−5,914,821 −5,861,391 0.0000000000000% −5,861,392 −5,807,9620.0000000000000% −5,807,962 −5,754,532 0.0000000000000% −5,754,533−5,701,103 0.0000000000000% −5,701,103 −5,647,673 0.0000000000000%−5,647,674 −5,594,244 0.0000000000000% −5,594,244 −5,540,8140.0000000000000% −5,540,815 −5,487,385 0.0000000000000% −5,487,385−5,433,955 0.0000000000000% −5,433,956 −5,380,526 0.0000000000000%−5,380,526 −5,327,096 0.0000000000000% −5,327,097 −5,273,6670.0000000000000% −5,273,667 −5,220,237 0.0000000000000% −5,220,238−5,166,808 0.0000000000000% −5,166,809 −5,113,379 0.0000000000000%−5,113,379 −5,059,949 0.0000000000000% −5,059,950 −5,006,5200.0000000000000% −5,006,520 −4,953,090 0.0000000000000% −4,953,091−4,899,661 0.0000000000000% −4,899,661 −4,846,231 0.0000000000000%−4,846,232 −4,792,802 0.0000000000000% −4,792,802 −4,739,3720.0000000000000% −4,739,373 −4,685,943 0.0000000000000% −4,685,943−4,632,513 0.0000000000000% −4,632,514 −4,579,084 0.0000000000000%−4,579,084 −4,525,654 0.0000000000000% −4,525,655 −4,472,2250.0000000000000% −4,472,225 −4,418,795 0.0000000000000% −4,418,796−4,365,366 0.0000000000000% −4,365,366 −4,311,936 0.0000000000000%−4,311,937 −4,258,507 0.0000000000000% −4,258,508 −4,205,0780.0000000000000% −4,205,078 −4,151,648 0.0000000000000% −4,151,649−4,098,219 0.0000000000000% −4,098,219 −4,044,789 0.0000000000000%−4,044,790 −3,991,360 0.0000000000000% −3,991,360 −3,937,9300.0000000000000% −3,937,931 −3,884,501 0.0000000000000% −3,884,501−3,831,071 0.0000000000000% −3,831,072 −3,777,642 0.0000000000000%−3,777,642 −3,724,212 0.0000000000000% −3,724,213 −3,670,7830.0000000000000% −3,670,783 −3,617,353 0.0000000000000% −3,617,354−3,563,924 0.0000000000000% −3,563,924 −3,510,494 0.0000000000000%−3,510,495 −3,457,065 0.0000000000000% −3,457,065 −3,403,6350.0000000000000% −3,403,636 −3,350,206 0.0000000000000% −3,350,206−3,296,776 0.0000000000000% −3,296,777 −3,243,347 0.0000000000000%−3,243,348 −3,189,918 0.0000000000000% −3,189,918 −3,136,4880.0000000000000% −3,136,489 −3,083,059 0.0000000000000% −3,083,059−3,029,629 0.0000000000000% −3,029,630 −2,976,200 0.0000000000000%−2,976,200 −2,922,770 0.0000000000000% −2,922,771 −2,869,3410.0000000000000% −2,869,341 −2,815,911 0.0000000000000% −2,815,912−2,762,482 0.0000000000002% −2,762,482 −2,709,052 0.0000000000008%−2,709,053 −2,655,623 0.0000000000033% −2,655,623 −2,602,1930.0000000000125% −2,602,194 −2,548,764 0.0000000000475% −2,548,764−2,495,334 0.0000000001747% −2,495,335 −2,441,905 0.0000000006651%−2,441,905 −2,388,475 0.0000000024238% −2,388,476 −2,335,0460.0000000080978% −2,335,047 −2,281,617 0.0000000253481% −2,281,617−2,228,187 0.0000000733211% −2,228,188 −2,174,758 0.0000001867159%−2,174,758 −2,121,328 0.0000004261562% −2,121,329 −2,067,8990.0000009313194% −2,067,899 −2,014,469 0.0000019335706% −2,014,470−1,961,040 0.0000038676092% −1,961,040 −1,907,610 0.0000074638267%−1,907,611 −1,854,181 0.0000141004062% −1,854,181 −1,800,7510.0000257187458% −1,800,752 −1,747,322 0.0000458625286% −1,747,322−1,693,892 0.0000795564894% −1,693,893 −1,640,463 0.0001345972909%−1,640,463 −1,587,033 0.0002246125068% −1,587,034 −1,533,6040.0003644253615% −1,533,604 −1,480,174 0.0005839457471% −1,480,175−1,426,745 0.0009214492554% −1,426,746 −1,373,316 0.0014287777387%−1,373,316 −1,319,886 0.0021861542251% −1,319,887 −1,266,4570.0032501470111% −1,266,457 −1,213,027 0.0047862710163% −1,213,028−1,159,598 0.0069321633908% −1,159,598 −1,106,168 0.0098108149165%−1,106,169 −1,052,739 0.0137710109755% −1,052,739 −999,3090.0191361811519% −999,310 −945,880 0.0263632903599% −945,880 −892,4500.0364035878949% −892,451 −839,021 0.0499307672471% −839,021 −785,5910.0683278672882% −785,592 −732,162 0.0933978212063% −732,162 −678,7320.1256591657555% −678,733 −625,303 0.1683302008600% −625,303 −571,8730.2216802362905% −571,874 −518,444 0.2902420118881% −518,444 −465,0140.3753515510399% −465,015 −411,585 0.4802162030795% −411,586 −358,1560.6150651589033% −358,156 −304,726 0.7776280285252% −304,727 −251,2970.9823703282698% −251,297 −197,867 1.2387139540238% −197,868 −144,4381.5505610613359% −144,438 −91,008 1.9345467317144% −91,009 −37,5792.3948077231803% −37,579 15,851 2.9538798318203% 15,850 69,2803.6179091831141% 69,280 122,710 4.3842791490843% 122,709 176,1395.2926808415697% 176,139 229,569 6.3307564485749% 229,568 282,9987.4937256652527% 282,998 336,428 8.8509768451619% 336,427 389,85710.3549745057132% 389,857 443,287 12.1026820733734% 443,286 496,71613.9865965070720% 496,715 550,145 16.1839804158823% 550,145 603,57518.6778992565306% 603,574 657,004 21.3131673703113% 657,004 710,43424.2901947774939% 710,433 763,863 27.4594989869027% 763,863 817,29330.8178257845057% 817,292 870,722 34.3156494106901% 870,722 924,15237.8993836918944% 924,151 977,581 41.7047034936096% 977,581 1,031,01145.4838918012672% 1,031,010 1,084,440 49.3518510412647% 1,084,4401,137,870 53.3242640527161% 1,137,869 1,191,299 57.2425970659684%1,191,299 1,244,729 61.2546072517691% 1,244,728 1,298,15865.0696980726194% 1,298,158 1,351,588 68.7239632498141% 1,351,5871,405,017 72.4133312265365% 1,405,017 1,458,447 75.7202577860827%1,458,446 1,511,876 78.8775420341887% 1,511,875 1,565,30581.6583654488451% 1,565,305 1,618,735 84.2768550319979% 1,618,7341,672,164 86.6393038496071% 1,672,164 1,725,594 88.6365901525135%1,725,593 1,779,023 90.5070476758832% 1,779,023 1,832,45392.0916604232132% 1,832,452 1,885,882 93.4520913732819% 1,885,8821,939,312 94.6641437596668% 1,939,311 1,992,741 95.6604823479376%1,992,741 2,046,171 96.5398453395601% 2,046,170 2,099,60097.2399787681478% 2,099,600 2,153,030 97.8256265208915% 2,153,0292,206,459 98.3203924612396% 2,206,459 2,259,889 98.7015365259410%2,259,888 2,313,318 99.0159661616815% 2,313,318 2,366,74899.2576908230848% 2,366,747 2,420,177 99.4452148724730% 2,420,1762,473,606 99.5928985523235% 2,473,606 2,527,036 99.6991870185069%2,527,035 2,580,465 99.7831157143216% 2,580,465 2,633,89599.8438233660748% 2,633,894 2,687,324 99.8886319332064% 2,687,3242,740,754 99.9218006251294% 2,740,753 2,794,183 99.9452612568851%2,794,183 2,847,613 99.9625774871492% 2,847,612 2,901,04299.9744035061454% 2,901,042 2,954,472 99.9827838751501% 2,954,4713,007,901 99.9886010922153% 3,007,901 3,061,331 99.9924671542066%3,061,330 3,114,760 99.9951488335340% 3,114,760 3,168,19099.9968908688022% 3,168,189 3,221,619 99.9980405960304% 3,221,6193,275,049 99.9987797593890% 3,275,048 3,328,478 99.9992410744171%3,328,477 3,381,907 99.9995434133531% 3,381,907 3,435,33799.9997257966347% 3,435,336 3,488,766 99.9998370151959% 3,488,7663,542,196 99.9999052533947% 3,542,195 3,595,625 99.9999451782640%3,595,625 3,649,055 99.9999689426822% 3,649,054 3,702,48499.9999826429842% 3,702,484 3,755,914 99.9999903458163% 3,755,9133,809,343 99.9999947534083% 3,809,343 3,862,773 99.9999971465665%3,862,772 3,916,202 99.9999985024046% 3,916,202 3,969,63299.9999992188081% 3,969,631 4,023,061 99.9999995999997% 4,023,0614,076,491 99.9999997988438% 4,076,490 4,129,920 99.9999998993262%4,129,920 4,183,350 99.9999999521190% 4,183,349 4,236,77999.9999999771398% 4,236,779 4,290,209 99.9999999894436% 4,290,2084,343,638 99.9999999951710% 4,343,637 4,397,067 99.9999999978285%4,397,067 4,450,497 99.9999999990703% 4,450,496 4,503,92699.9999999995951% 4,503,926 4,557,356 99.9999999998311% 4,557,3554,610,785 99.9999999999319% 4,610,785 4,664,215 99.9999999999728%4,664,214 4,717,644 99.9999999999896% 4,717,644 4,771,07499.9999999999959% 4,771,073 4,824,503 99.9999999999985% 4,824,5034,877,933 99.9999999999995% 4,877,932 4,931,362 99.9999999999998%4,931,362 4,984,792 100.0000000000000% 4,984,791 5,038,221100.0000000000000% 5,038,221 5,091,651 100.0000000000000% 5,091,6505,145,080 100.0000000000000% 5,145,080 5,198,510 100.0000000000000%5,198,509 5,251,939 100.0000000000000% 5,251,938 5,305,368100.0000000000000% 5,305,368 5,358,798 100.0000000000000% 5,358,7975,412,227 100.0000000000000% 5,412,227 5,465,657 100.0000000000000%5,465,656 5,519,086 100.0000000000000% 5,519,086 5,572,516100.0000000000000% 5,572,515 5,625,945 100.0000000000000% 5,625,9455,679,375 100.0000000000000% 5,679,374 5,732,804 100.0000000000000%5,732,804 5,786,234 100.0000000000000% 5,786,233 5,839,663100.0000000000000% 5,839,663 5,893,093 100.0000000000000% 5,893,0925,946,522 100.0000000000000% 5,946,522 5,999,952 100.0000000000000%5,999,951 6,053,381 100.0000000000000% 6,053,381 6,106,811100.0000000000000% 6,106,810 6,160,240 100.0000000000000% 6,160,2406,213,670 100.0000000000000% 6,213,669 6,267,099 100.0000000000000%6,267,098 6,320,528 100.0000000000000% 6,320,528 6,373,958100.0000000000000% 6,373,957 6,427,387 100.0000000000000% 6,427,3876,480,817 100.0000000000000% 6,480,816 6,534,246 100.0000000000000%6,534,246 6,587,676 100.0000000000000% 6,587,675 6,641,105100.0000000000000%

1. A method for constructing a historically based frequency distributionof unknown ultimate outcomes in a data set, the method comprising thefollowing acts: A. collecting relevant data about a series of knowncohorts, where a new group of the data emerges at regular timeintervals, measuring a characteristic of each group of the data atregular time intervals, and entering each said characteristic into adata set having at least two dimensions; B. determining a number offrequency intervals N to be used to construct said distribution of uownultimate outcomes; C. for each period I, constructing an aggregatedistribution by: (a) calculating period-to-period ratios of the datacharacteristics; (b) identifing a range of ratio outcomes for cohort I;(c) constructing subintervals for cohort I; and (d) calculating allpossible ratio outcomes for cohort I; (e) inserting each outcome intothe proper interval; and D. constructing a convolution distribution ofoutcomes (said historically based frequency distribution of unknownultimate outcomes) for all said possible ratio cohorts combined, by: (a)selecting outcomes for any two cohorts A and B; (b) constructing a newrange of outcomes for the convolution distribution of cohorts A and B;(c) constructing new subintervals for the convolution distribution ofcohorts A and B; (d) calculating the combined outcomes for the twocohorts A and B to provide a resulting convolution distribution; and (e)combining the resulting convolution distribution with the distributionof outcomes for each remaining cohort by repeating each of the precedingacts D.(a) through D.(d) for each pair of cohorts.
 2. The method ofclaim 1, in which N is a number of intervals required to meet a givenlevel of error tolerance selected by a user.
 3. The method of claim 1,in which N is a maximum number of intervals that can be calculated by acomputer provided by a user in a given period of time.
 4. The method ofclaim 1, futher comprising the acts of (a) constructing convolutiondistributions for at least two separate groups of data using the methoddescribed in claim 1; and (b) constructing a convolution distribution ofsuch separate groups together.
 5. A computer software system having aset of instructions for controlling a general purpose digital computerin performing a reserve measure function comprising: a set ofinstructions for: A. receiving a set of data, B. receiving a number ofintervals N, C. for each period I, constructing the aggregatedistribution by: (a) calculating the period-to-period ratios of thedata; (b) identifyg a range of ratio outcomes for cohort I; (c)constructing subintervals for cohort I; (d) calculating all possibleratio outcomes for cohort I; (e) inserting each outcome into the properinterval; and D. constructing a convolution distribution for all saidpossible ratio cohorts combined, by: (a) selecting outcomes for any twocohorts A and B; (b) constructing a new range of ratio outcomes for theconvolution distribution of cohorts A and B; (c) constructing newsubintervals for the convolution distribution of cohorts A and B; (d)calculating the combined possible ratio outcomes for the two cohorts Aand B; and (e) combining the resulting convolution distribution with thedistribution of outcomes for each remaining cohort by repeating each ofthe preceding actions D.(a) through D.(d) for constructing a newconvolution distribution.
 6. The computer software system of claim 5,where N is a number of intervals required to meet a given level of errortolerance as determined by a user.
 7. The computer software system ofclaim 5, further comprising a set of instructions for: receiving anerror tolerance ε selected by a user; calculating the number ofintervals N required to produce such level of error tolerance.
 8. Thecomputer software system of claim 5, in which N is a maximum number ofintervals that can be calculated by the computer in a given period oftime.
 9. The computer software system of claim 5, in which a value for Nis fixed in the instructions.
 10. The computer software system of claim5, in which N is a number selected by a user.
 11. The computer softwaresystem of claim 5, in which the set of data is comprised of insuredlosses over a given period of years and for a given line of businesses.12. A computer-readable medium storing instructions executable by acomputer to cause the computer to perform a reserve measure processcomprising: A. receiving a set of data; B. receiving a number ofintervals N; C. for each period I, constructing the aggregatedistribution by: (a) calculating the period-to-period ratios; (b)identifying the range of outcomes for cohort I; (c) constructing thesubintervals for cohort I; and (d) calculating all the differentoutcomes for cohort I (e) inserting each outcome mto the properinterval; and D. constructing a convolution distribution for all cohortscombined, by: (a) selecting any two cohorts A and B (b) constructing anew range of outcomes for the convolution distribution of cohorts A andB; (c) constructing new subintervals for the convolution distribution ofcohorts A and B; (d) calculating the combined outcomes for the twocohorts A and B; and (e) combining the resulting convolutiondistribution with the distribution of outcomes for each remaining cohortby repeating each of the preceding actions D.(a) through D.(d) forconstructing a new convolution distribution.
 13. The computer readablemedium of instructions of claim 12, where N is the number of intervalsrequired to meet a given level of error tolerance as determined by theuser.
 14. The computer readable medium of instructions of claim 12,further comprising a set of instructions for: receiving an errortolerance ε selected by the user; calculating the number of intervals Nrequired to produce such level of error tolerance.
 15. The computerreadable medium of instructions of claim 12, in which N is the maximumnumber of intervals that can be calculated by the computer in a givenperiod of time.
 16. The computer readable medium of instructions ofclaim 12, in which a value for N is fixed in the instructions.
 17. Thecomputer readable medium of instructions of claim 12, in which N is anumber selected by the user.
 18. The computer readable medium ofinstructions of claim 12, in which the data set is comprised of insuredlosses over a given period of years and for a given line of businesses.19. A method for constructing a historically based frequencydistribution of insurance losses, the method comprising the followingacts: A. collection of relevant data about claims experience across aline of businesses, for a set of accident years; B. determination of anumber of intervals N to be used to construct said distribution ofinsurance losses; C. for each accident year I in each line of businessK, constructing the aggregate distribution by: (a) calculating theperiod-to-period ratios; (b) identifying the range of outcomes foraccident year I; (c) constructing the subintervals for accident year I;(d) calculating all the different outcomes for accident year I (e)inserting each outcome into the proper interval; and; D. for each lineof business K, constructing a convolution distribution for all accidentyears combined, by: (a) selecting any two accident years A and B; (b)constructing a new range of outcomes for the convolution distribution ofaccident years A and B; (c) constructing new subintervals for theconvolution distribution of accident years A and B; (d) calculating thecombined outcomes for the two accident years A and B; (e) combining theresulting convolution distribution with the distribution of outcomes foreach remaining accident year by repeating each of the preceding stepsD.(a) through D.(d) for constructing a new convolution distribution; andF. combining the resultant convolution distributions for all lines ofbusiness by (a) selecting any two lines of business X and Y; (b)constructing a new range of outcomes for the convolution distribution oflines of business X and Y; (c) construcing new subintervals for theconvolution distribution of lines of business X and Y; (d) calculatingthe combined outcomes for the two lines of business X and Y; and (e)combing the resulting convolution distribution with the disribution ofoutcomes for each remaining line of business by repeating each of thepreceding steps F.(a) through F.(d) for constructing a new convolutiondistribution to produce a convolution distribution across all lines ofbusiness.
 20. The method of claim 19, further comprising the followingaction: evaluating the actual insurance reserve based on the resultingconvolution distribution.
 21. The method of claim 20, further comprisingthe following action: adjusting the insurance reserve of the user basedupon the comparison of the actual reserve to the convolutiondistribution.
 22. The method of claim 19, further comprising thefollowing action: selecting an insurance reserve based upon theresulting convolution distribution.